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1
Abdullah, Ibrahima Faye, and Laila Amera Aziz. "Artificial Neural Networks Solutions for Solving Differential Equations: A Focus and Example for Flow of Viscoelastic Fluid with Microrotation." Journal of Advanced Research in Fluid Mechanics and Thermal Sciences 112, no. 1 (January 13, 2024): 76–83. http://dx.doi.org/10.37934/arfmts.112.1.7683.
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Physics-informed neural networks (PINN) are an artificial neural network (ANN) approach for solving differential equations. PINN offers an alternative to classical numerical methods. The paper discusses the applications of PINN in various domains by highlighting the advantages, challenges, limitations, and some future directions. For example, PINN is implemented to solve the differential equations describing the Flow of Viscoelastic Fluid with Microrotation at a Horizontal Circular Cylinder Boundary Layer. The differential equations resulting from a nondimensionalization process of the governing equations and the associated boundary conditions are solved using PINN. The obtained results using PINN are discussed and compared to other state-of-the-art methods. Future research might aim to increase the precision and effectiveness of PINN models for solving differential equations, either by adding more physics-based restrictions or multi-scale methods to expand their capabilities. Additionally, investigating new application domains like linked multi-physics issues or real-time simulation situations may help to increase the reach and significance of PINN approaches.
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Zhang, Wenjuan, and Mohammed Al Kobaisi. "On the Monotonicity and Positivity of Physics-Informed Neural Networks for Highly Anisotropic Diffusion Equations." Energies 15, no. 18 (September 18, 2022): 6823. http://dx.doi.org/10.3390/en15186823.
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Physics-informed neural network (PINN) models are developed in this work for solving highly anisotropic diffusion equations. Compared to traditional numerical discretization schemes such as the finite volume method and finite element method, PINN models are meshless and, therefore, have the advantage of imposing no constraint on the orientations of the diffusion tensors or the grid orthogonality conditions. To impose solution positivity, we tested PINN models with positivity-preserving activation functions for the last layer and found that the accuracy of the corresponding PINN solutions is quite poor compared to the vanilla PINN model. Therefore, to improve the monotonicity properties of PINN models, we propose a new loss function that incorporates additional terms which penalize negative solutions, in addition to the usual partial differential equation (PDE) residuals and boundary mismatch. Various numerical experiments show that the PINN models can accurately capture the tensorial effect of the diffusion tensor, and the PINN model utilizing the new loss function can reduce the degree of violations of monotonicity and improve the accuracy of solutions compared to the vanilla PINN model, while the computational expenses remain comparable. Moreover, we further developed PINN models that are composed of multiple neural networks to deal with discontinuous diffusion tensors. Pressure and flux continuity conditions on the discontinuity line are used to stitch the multiple networks into a single model by adding another loss term in the loss function. The resulting PINN models were shown to successfully solve the diffusion equation when the principal directions of the diffusion tensor change abruptly across the discontinuity line. The results demonstrate that the PINN models represent an attractive option for solving difficult anisotropic diffusion problems compared to traditional numerical discretization methods.
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Ang, Elijah Hao Wei, Guangjian Wang, and Bing Feng Ng. "Physics-Informed Neural Networks for Low Reynolds Number Flows over Cylinder." Energies 16, no. 12 (June 7, 2023): 4558. http://dx.doi.org/10.3390/en16124558.
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Physics-informed neural network (PINN) architectures are recent developments that can act as surrogate models for fluid dynamics in order to reduce computational costs. PINNs make use of deep neural networks, where the Navier-Stokes equation and freestream boundary conditions are used as losses of the neural network; hence, no simulation or experimental data in the training of the PINN is required. Here, the formulation of PINN for fluid dynamics is demonstrated and critical factors influencing the PINN design are discussed through a low Reynolds number flow over a cylinder. The PINN architecture showed the greatest improvement to the accuracy of results from the increase in the number of layers, followed by the increase in the number of points in the point cloud. Increasing the number of nodes per hidden layer brings about the smallest improvement in performance. In general, PINN is much more efficient than computational fluid dynamics (CFD) in terms of memory resource usage, with PINN requiring 5–10 times less memory. The tradeoff for this advantage is that it requires longer computational time, with PINN requiring approximately 3 times more than that of CFD. In essence, this paper demonstrates the direct formulation of PINN without the need for data, alongside hyperparameter design and comparison of computational requirements.
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Ekramoddoullah, Abul K. M., Doug Taylor, and Barbara J. Hawkins. "Characterisation of a fall protein of sugar pine and detection of its homologue associated with frost hardiness of western white pine needles." Canadian Journal of Forest Research 25, no. 7 (July 1, 1995): 1137–47. http://dx.doi.org/10.1139/x95-126.
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A 19-kDa protein of sugar pine (named Pinl I; i.e., protein I of Pinuslambertiana Dougl.) was detected in increasing amounts in the fall. This protein was separated by SDS–PAGE and also by two-dimensional gel electrophoresis. Pinl I was composed of two acidic isoforms. This protein was subjected to N-terminal amino acid sequence analysis. The two isoforms had an identical N-terminal amino acid sequence. The N-terminal peptide was synthesized and purified, and the purity of the synthetic peptide was greater than 95%. The peptide was conjugated to a carrier protein, keyhole limpet hemocyanin (KLH). Rabbits were immunized with peptide–KLH and the antipeptide antibody was purified from the crude antisera by immunoaffinity chromatography. The antibody was shown to bind specifically to PinI I. This anti-Pin I I antibody was used in a Western immunoblot to detect the homologues of Pin1 1 in two other five-needled pines: western white pine (Pinusmonticola Dougl.; named Pinm III) and eastern white pine (Pinusstrobus L.). The antibody was also used to monitor the seasonal variation of Pinm III in western white pine needles. Pinm III was shown to be associated with overwintering of western white pine seedlings. A significant correlation was observed between the frost hardiness of western white pine foliage and the content of Pinm III.
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Li, Jianfeng, Liangying Zhou, Jingwei Sun, and Guangzhong Sun. "Physically plausible and conservative solutions to Navier-Stokes equations using Physics-Informed CNNs." JUSTC 53 (2023): 1. http://dx.doi.org/10.52396/justc-2022-0174.
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Physics-informed Neural Network (PINN) is an emerging approach for efficiently solving partial differential equations (PDEs) using neural networks. Physics-informed Convolutional Neural Network (PICNN), a variant of PINN enhanced by convolutional neural networks (CNNs), has achieved better results on a series of PDEs since the parameter-sharing property of CNNs is effective to learn spatial dependencies. However, applying existing PICNN-based methods to solve Navier-Stokes equations can generate oscillating predictions, which are inconsistent with the laws of physics and the conservation properties. To address this issue, we propose a novel method that combines PICNN with the finite volume method to obtain physically plausible and conservative solutions to Navier-Stokes equations. We derive the second-order upwind difference scheme of Navier-Stokes equations using the finite volume method. Then we use the derived scheme to calculate the partial derivatives and construct the physics-informed loss function. The proposed method is assessed by experiments on steady-state Navier-Stokes equations under different scenarios, including convective heat transfer, lid-driven cavity flow, etc. The experimental results demonstrate that our method can effectively improve the plausibility and the accuracy of the predicted solutions from PICNN.
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Xiao, Zixu, Yaping Ju, Zhen Li, Jiawang Zhang, and Chuhua Zhang. "On the Hard Boundary Constraint Method for Fluid Flow Prediction based on the Physics-Informed Neural Network." Applied Sciences 14, no. 2 (January 19, 2024): 859. http://dx.doi.org/10.3390/app14020859.
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With the rapid development of artificial intelligence technology, the physics-informed neural network (PINN) has gradually emerged as an effective and potential method for solving N-S equations. The treatment of constraints is vital to the PINN prediction accuracy. Compared to soft constraints, hard constraints are advantageous for the avoidance of difficulties in guaranteeing definite conditions and determining penalty coefficients. However, the principles on the formulation of hard constraints of PINN currently remain to be formed, which hinders the application of PINN in engineering fields. In this study, hard-constraint-based PINN models are constructed for Couette flow, plate shear flow and stenotic/aneurysmal flow with curved geometries. Particular efforts have been devoted to assessing the impact of the model parameters of hard constraints, i.e., degree and scaling factor, on the prediction accuracy of PINN at different Reynolds numbers. The results show that the degree is the most important factor that influences the prediction accuracy, followed by the scaling factor. As for the N-S equations, the degree of hard constraints should be at least two, while the scaling factor is recommended to be maintained around 1.0. The outcomes of the present work are of reference value for the development of PINN methods in fluid mechanics.
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Xia, Yichun, and Yonggang Meng. "Physics-Informed Neural Network (PINN) for Solving Frictional Contact Temperature and Inversely Evaluating Relevant Input Parameters." Lubricants 12, no. 2 (February 17, 2024): 62. http://dx.doi.org/10.3390/lubricants12020062.
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Ensuring precise prediction, monitoring, and control of frictional contact temperature is imperative for the design and operation of advanced equipment. Currently, the measurement of frictional contact temperature remains a formidable challenge, while the accuracy of simulation results from conventional numerical methods remains uncertain. In this study, a PINN model that incorporates physical information, such as partial differential equation (PDE) and boundary conditions, into neural networks is proposed to solve forward and inverse problems of frictional contact temperature. Compared to the traditional numerical calculation method, the preprocessing of the PINN is more convenient. Another noteworthy characteristic of the PINN is that it can combine data to obtain a more accurate temperature field and solve inverse problems to identify some unknown parameters. The experimental results substantiate that the PINN effectively resolves the forward problems of frictional contact temperature when provided with known input conditions. Additionally, the PINN demonstrates its ability to accurately predict the friction temperature field with an unknown input parameter, which is achieved by incorporating a limited quantity of easily measurable actual temperature data. The PINN can also be employed for the inverse identification of unknown parameters. Finally, the PINN exhibits potential in solving inverse problems associated with frictional contact temperature, even when multiple input parameters are unknown.
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Chen, Yanlai, Yajie Ji, Akil Narayan, and Zhenli Xu. "TGPT-PINN: Nonlinear model reduction with transformed GPT-PINNs." Computer Methods in Applied Mechanics and Engineering 430 (October 2024): 117198. http://dx.doi.org/10.1016/j.cma.2024.117198.
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Ngo, Son Ich, and Young-Il Lim. "Solution and Parameter Identification of a Fixed-Bed Reactor Model for Catalytic CO2 Methanation Using Physics-Informed Neural Networks." Catalysts 11, no. 11 (October 28, 2021): 1304. http://dx.doi.org/10.3390/catal11111304.
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In this study, we develop physics-informed neural networks (PINNs) to solve an isothermal fixed-bed (IFB) model for catalytic CO2 methanation. The PINN includes a feed-forward artificial neural network (FF-ANN) and physics-informed constraints, such as governing equations, boundary conditions, and reaction kinetics. The most effective PINN structure consists of 5–7 hidden layers, 256 neurons per layer, and a hyperbolic tangent (tanh) activation function. The forward PINN model solves the plug-flow reactor model of the IFB, whereas the inverse PINN model reveals an unknown effectiveness factor involved in the reaction kinetics. The forward PINN shows excellent extrapolation performance with an accuracy of 88.1% when concentrations outside the training domain are predicted using only one-sixth of the entire domain. The inverse PINN model identifies an unknown effectiveness factor with an error of 0.3%, even for a small number of observation datasets (e.g., 20 sets). These results suggest that forward and inverse PINNs can be used in the solution and system identification of fixed-bed models with chemical reaction kinetics.
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Hou, Qingzhi, Honghan Du, Zewei Sun, Jianping Wang, Xiaojing Wang, and Jianguo Wei. "PINN-CDR: A Neural Network-Based Simulation Tool for Convection-Diffusion-Reaction Systems." International Journal of Intelligent Systems 2023 (August 16, 2023): 1–15. http://dx.doi.org/10.1155/2023/2973249.
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In this paper, a discretization-free approach based on the physics-informed neural network (PINN) is proposed for solving the forward and inverse problems governed by the nonlinear convection-diffusion-reaction (CDR) systems. By embedding physical information described by the CDR system in the feedforward neural networks, PINN is trained to approximate the solution of the system without the need of labeled data. The good performance of PINN in solving the forward problem of the nonlinear CDR systems is verified by studying the problems of gas-solid adsorption and autocatalytic reacting flow. For CDR systems with different Péclet number, PINN can largely eliminate the numerical diffusion and unphysical oscillations in traditional numerical methods caused by high Péclet number. Meanwhile, the PINN framework is implemented to solve the inverse problem of nonlinear CDR systems and the results show that the unknown parameters can be effectively recognized even with high noisy data. It is concluded that the established PINN algorithm has good accuracy, convergence, and robustness for both the forward and inverse problems of CDR systems.
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Pu, Jun, Wenfang Song, Junlai Wu, Feifei Gou, Xia Yin, and Yunqian Long. "PINN-Based Method for Predicting Flow Field Distribution of the Tight Reservoir after Fracturing." Geofluids 2022 (April 6, 2022): 1–10. http://dx.doi.org/10.1155/2022/1781388.
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The physical-informed neural network (PINN) model can greatly improve the ability to fit nonlinear data with the incorporation of prior knowledge, which endows traditional neural networks with interpretability. Considering the seepage law in the tight reservoir after hydraulic fracturing, a model based on PINN and two-dimensional seepage physical equations was proposed, which can effectively predict the flow field distribution of the tight reservoir after fracturing. Firstly, the dataset was obtained based on physical and numerical models of the tight reservoirs developed by volume fracturing. Furthermore, coupling the neural networks and the two-dimensional unsteady seepage equation, a PINN model was developed to predict the flow field distribution of the tight reservoir. Finally, a systematic study was performed concerning the noise corruption levels, training iterations, and training sample size that affect the prediction results of PINN models. Besides, a comparison between PINN and traditional deep neural networks (DNN) was presented. The results show that the DNN model was not only sensitive to noisy data but also more vulnerable to overfitting as the training iterations increase. In addition, the prediction accuracy cannot be guaranteed when the samples are inadequate (<500). In contrast, the PINN model was less affected by noise and training iterations and thus indicates greater stability. Moreover, the PINN model outperforms the DNN model in the case of inadequate samples attributing to prior knowledge. This study confirms that the adopted PINN model can provide algorithmic support for the accurate prediction of flow field distribution of the tight reservoirs.
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Vanden Berge, Dennis J., Nicholas A. Kusnezov, Sydney Rubin, Thomas Dagg, Justin Orr, Justin Mitchell, Miguel Pirela-Cruz, and John C. Dunn. "Outcomes Following Isolated Posterior Interosseous Nerve Neurectomy: A Systematic Review." HAND 12, no. 6 (February 1, 2017): 535–40. http://dx.doi.org/10.1177/1558944717692093.
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Background: Posterior interosseous nerve neurectomies (PINN) are an option in the treatment of chronic dorsal wrist pain. However, the literature describing PINN consists primarily of small case series, and the procedure is typically done as an adjunct treatment; therefore, the outcomes of the PINN itself are not well known. We performed a systematic review of the literature to provide characteristics of patients following a PINN. Methods: A systematic review of the literature was performed. Papers published in the PubMed database in English on isolated PINN were included. Articles in which a PINN was performed as an adjunct were excluded. Primary outcomes were return to work, patient satisfaction, pain/function scores, wrist range of motion, complications, and pain recurrence. Weighted averages were used to calculate continuous data, whereas categorical data were noted in percentages. Results: The search yielded 427 articles including 6 studies and 135 patients (136 cases). The average age was 43.6 years (range, 17-75), and most patients were female (54.1%). At an average final follow-up of 51 months, 88.9% of patients were able to return to work. After initial improvement, a recurrence of pain occurred in 25.5% of patients at an average of 12.3 months. Excluding recurrence of pain, the complication rate was 0.9%, including 1 reflex sympathetic dystrophy. Overall, 88.4% of patients experienced a subjective improvement and were satisfied with the procedure. Conclusions: Isolated PINN have shown excellent clinical outcomes, with few patients experiencing recurrent pain at long-term follow-up. PINN can provide relief in patient’s chronic wrist pain.
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Ta, Hoa, Shi Wen Wong, Nathan McClanahan, Jung-Han Kimn, and Kaiqun Fu. "Exploration on Physics-Informed Neural Networks on Partial Differential Equations (Student Abstract)." Proceedings of the AAAI Conference on Artificial Intelligence 37, no. 13 (June 26, 2023): 16344–45. http://dx.doi.org/10.1609/aaai.v37i13.27032.
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Data-driven related solutions are dominating various scientific fields with the assistance of machine learning and data analytics. Finding effective solutions has long been discussed in the area of machine learning. The recent decade has witnessed the promising performance of the Physics-Informed Neural Networks (PINN) in bridging the gap between real-world scientific problems and machine learning models. In this paper, we explore the behavior of PINN in a particular range of different diffusion coefficients under specific boundary conditions. In addition, different initial conditions of partial differential equations are solved by applying the proposed PINN. Our paper illustrates how the effectiveness of the PINN can change under various scenarios. As a result, we demonstrate a better insight into the behaviors of the PINN and how to make the proposed method more robust while encountering different scientific and engineering problems.
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Park, Hyun-Woo, and Jin-Ho Hwang. "Predicting the Early-Age Time-Dependent Behaviors of a Prestressed Concrete Beam by Using Physics-Informed Neural Network." Sensors 23, no. 14 (July 24, 2023): 6649. http://dx.doi.org/10.3390/s23146649.
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This paper proposes a physics-informed neural network (PINN) for predicting the early-age time-dependent behaviors of prestressed concrete beams. The PINN utilizes deep neural networks to learn the time-dependent coupling among the effective prestress force and the several factors that affect the time-dependent behavior of the beam, such as concrete creep and shrinkage, tendon relaxation, and changes in concrete elastic modulus. Unlike traditional numerical algorithms such as the finite difference method, the PINN directly solves the integro-differential equation without the need for discretization, offering an efficient and accurate solution. Considering the trade-off between solution accuracy and the computing cost, optimal hyperparameter combinations are determined for the PINN. The proposed PINN is verified through the comparison to the numerical results from the finite difference method for two representative cross sections of PSC beams.
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Pratama, Muchamad Harry Yudha, and Agus Yodi Gunawan. "EXPLORING PHYSICS-INFORMED NEURAL NETWORKS FOR SOLVING BOUNDARY LAYER PROBLEMS." Journal of Fundamental Mathematics and Applications (JFMA) 6, no. 2 (November 27, 2023): 101–16. http://dx.doi.org/10.14710/jfma.v6i2.20084.
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In this paper, we explore a cutting-edge technique called as Physics- Informed Neural Networks (PINN) to tackle boundary layer problems. We here examine four different cases of boundary layers of second-order ODE: a linear ODEwith constant coefficients, a nonlinear ODE with homogeneous boundary conditions, an ODE with non-constant coefficients, and an ODE featuring multiple boundary layers. We adapt the line of PINN technique for handling those problems, and our results show that the accuracy of the resulted solutions depends on how we choose the most reliable and robust activation functions when designing the architecture of the PINN. Beside that, through our explorations, we aim to improve our understanding on how the PINN technique works better for boundary layer problems. Especially, the use of the SiLU (Sigmoid-Weighted Linear Unit) activation function in PINN has proven to be particularly remarkable in handling our boundary layer problems.
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Zhou, Meijun, and Gang Mei. "Transfer Learning-Based Coupling of Smoothed Finite Element Method and Physics-Informed Neural Network for Solving Elastoplastic Inverse Problems." Mathematics 11, no. 11 (May 31, 2023): 2529. http://dx.doi.org/10.3390/math11112529.
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In practical engineering applications, there is a high demand for inverting parameters for various materials, and obtaining monitoring data can be costly. Traditional inverse methods often involve tedious computational processes, require significant computational effort, and exhibit slow convergence speeds. The recently proposed Physics-Informed Neural Network (PINN) has shown great potential in solving inverse problems. Therefore, in this paper, we propose a transfer learning-based coupling of the Smoothed Finite Element Method (S-FEM) and PINN methods for the inversion of parameters in elastic-plasticity problems. The aim is to improve the accuracy and efficiency of parameter inversion for different elastic-plastic materials with limited data. High-quality small datasets were synthesized using S-FEM and subsequently combined with PINN for pre-training purposes. The parameters of the pre-trained model were saved and used as the initial state for the PINN model in the inversion of new material parameters. The inversion performance of the coupling of S-FEM and PINN is compared with the coupling of the conventional Finite Element Method (FEM) and PINN on a small data set. Additionally, we compared the efficiency and accuracy of both the transfer learning-based and non-transfer learning-based methods of the coupling of S-FEM and PINN in the inversion of different material parameters. The results show that: (1) our method performs well on small datasets, with an inversion error of essentially less than 2%; (2) our approach outperforms the coupling of conventional FEM and PINN in terms of both computational accuracy and computational efficiency; and (3) our approach is at least twice as efficient as the coupling of S-FEM and PINN without transfer learning, while still maintaining accuracy. Our method is well-suited for the inversion of different material parameters using only small datasets. The use of transfer learning greatly improves computational efficiency, making our method an efficient and accurate solution for reducing computational cost and complexity in practical engineering applications.
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Liu, Youqiong, Li Cai, Yaping Chen, and Bin Wang. "Physics-informed neural networks based on adaptive weighted loss functions for Hamilton-Jacobi equations." Mathematical Biosciences and Engineering 19, no. 12 (2022): 12866–96. http://dx.doi.org/10.3934/mbe.2022601.
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<abstract><p>Physics-informed neural networks (PINN) have lately become a research hotspot in the interdisciplinary field of machine learning and computational mathematics thanks to the flexibility in tackling forward and inverse problems. In this work, we explore the generality of the PINN training algorithm for solving Hamilton-Jacobi equations, and propose physics-informed neural networks based on adaptive weighted loss functions (AW-PINN) that is trained to solve unsupervised learning tasks with fewer training data while physical information constraints are imposed during the training process. To balance the contributions from different constrains automatically, the AW-PINN training algorithm adaptively update the weight coefficients of different loss terms by using the logarithmic mean to avoid additional hyperparameter. Moreover, the proposed AW-PINN algorithm imposes the periodicity requirement on the boundary condition and its gradient. The fully connected feedforward neural networks are considered and the optimizing procedure is taken as the Adam optimizer for some steps followed by the L-BFGS-B optimizer. The series of numerical experiments illustrate that the proposed algorithm effectively achieves noticeable improvements in predictive accuracy and the convergence rate of the total training error, and can approximate the solution even when the Hamiltonian is nonconvex. A comparison between the proposed algorithm and the original PINN algorithm for Hamilton-Jacobi equations indicates that the proposed AW-PINN algorithm can train the solutions more accurately with fewer iterations.</p></abstract>
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Fu, Cifu, Jie Xiong, and Fujiang Yu. "Storm surge forecasting based on physics-informed neural networks in the Bohai Sea." Journal of Physics: Conference Series 2718, no. 1 (March 1, 2024): 012057. http://dx.doi.org/10.1088/1742-6596/2718/1/012057.
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Abstract Physics-informed neural networks (PINN), as a new method of integrating artificial neural networks (ANN) and physical laws, have been considered and applied in the fields of ocean forecasting and ocean research. In this paper, the simplified two-dimensional (2D) storm surge governing equation is introduced into an ANN to establish a PINN-based storm surge forecast model. The numerical simulation results of 14 storm surge events in the Bohai Sea are selected as the PINN training set, and 6.3% of the training set data are randomly selected to reconstruct the storm surge field information. The storm surge reconstructed at each tide station is nearly identical to the storm surge curve simulated by the numerical model, with the root mean square error (RMSE) less than 0.12 m and absolute error of maximum storm surge less than 0.2 m. The analysis of the storm surge field at key moments (storm surge height lager than 1 m) shows that the difference in storm surge field between the PINN reconstruction and the numerical model is generally less than 0.4 m. Two storm surge events in the Bohai Sea are selected as forecast cases, and the same network structure, parameters, and storm surge data assimilation scheme are used for predictions by the ANN, PINN, and numerical model. The results show that compared to the ANN and numerical models, the average relative error of the maximum storm surge predicted by the PINN is reduced by approximately 25%, which significantly improves the forecast accuracy, therefore, the PINN is suitable for storm surge forecasting and research due to its advantages in small sample data training and strong physical meaning.
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Garcia Inda, Adan Jafet, Shao Ying Huang, Nevrez İmamoğlu, Ruian Qin, Tianyi Yang, Tiao Chen, Zilong Yuan, and Wenwei Yu. "Physics Informed Neural Networks (PINN) for Low Snr Magnetic Resonance Electrical Properties Tomography (MREPT)." Diagnostics 12, no. 11 (October 29, 2022): 2627. http://dx.doi.org/10.3390/diagnostics12112627.
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Electrical properties (EPs) of tissues facilitate early detection of cancerous tissues. Magnetic resonance electrical properties tomography (MREPT) is a technique to non-invasively probe the EPs of tissues from MRI measurements. Most MREPT methods rely on numerical differentiation (ND) to solve partial differential Equations (PDEs) to reconstruct the EPs. However, they are not practical for clinical data because ND is noise sensitive and the MRI measurements for MREPT are noisy in nature. Recently, Physics informed neural networks (PINNs) have been introduced to solve PDEs by substituting ND with automatic differentiation (AD). To the best of our knowledge, it has not been applied to MREPT due to the challenges in using PINN on MREPT as (i) a PINN requires part of ground-truth EPs as collocation points to optimize the network’s AD, (ii) the noisy input data disrupts the optimization of PINNs despite the noise-filtering nature of NNs and additional denoising processes. In this work, we propose a PINN-MREPT model based on a canonical analytic MREPT model. A reference padding layer with known EPs was added to surround the region of interest for providing additive collocation points. Moreover, an optimizable diffusion coefficient was embedded in the analytic MREPT model used in the PINN-MREPT. The noise robustness of the proposed PINN-MREPT for single-sample reconstruction was tested by using numerical phantoms of human brain with extra tumor-like tissues at different noise levels. The results of numerical experiments show that PINN-MREPT outperforms two typical numerical MREPT methods in terms of reconstruction accuracy, sensitivity to the extra tissues, and the correlations of line profiles in the regions of interest. The advantage of the PINN-MREPT is shown by the results of an experiment on phantom measurement, too. Moreover, it is found that the diffusion term plays an important role to achieve a noise-robust PINN-MREPT. This is an important step moving forward to a clinical application of MREPT.
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Lee, Jonghwan. "Physics-Informed Neural Network for High Frequency Noise Performance in Quasi-Ballistic MOSFETs." Electronics 10, no. 18 (September 10, 2021): 2219. http://dx.doi.org/10.3390/electronics10182219.
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A physics-informed neural network (PINN) model is presented to predict the nonlinear characteristics of high frequency (HF) noise performance in quasi-ballistic MOSFETs. The PINN model is formulated by combining the radial basis function-artificial neural networks (RBF-ANNs) with an improved noise equivalent circuit model, including all the noise sources. The RBF-ANNs are utilized to model the thermal channel noise, induced gate noise, correlation noise, as well as the shot noise, due to the gate and source-drain tunneling current through the potential barriers. By training a spatial distribution of the thermal channel noise and a Fano factor of the shot noise, underlying physical theories are naturally embedded into the PINN model as prior information. The PINN model shows good capability of predicting the noise performance at high frequencies.
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Lee, Sangmin, and John Popovics. "Applications of physics-informed neural networks for property characterization of complex materials." RILEM Technical Letters 7 (January 30, 2023): 178–88. http://dx.doi.org/10.21809/rilemtechlett.2022.174.
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The characterization of in-place material properties is important for quality control and condition assessment of the built infrastructure. Although various methods have been developed to characterize structural materials in situ, many suffer limitations and cannot provide complete or desired characterization, especially for inhomogeneous and complex materials such as concrete and rock. Recent advances in machine learning and artificial neural networks (ANN) can help address these limitations. In particular, physics-informed neural networks (PINN) portend notable advantages over traditional physics-based or purely data-driven approaches. PINN is a particular form of ANN, where physics-based equations are embedded within an ANN structure in order to regularize the outputs during the training process. This paper reviews the fundamentals of PINN, notes its differences from traditional ANN, and reviews applications of PINN for selected material characterization tasks. A specific application example is presented where mechanical wave propagation data are used to characterize in-place material properties. Ultrasonic data are obtained from experiments on long rod-shaped mortar and glass samples; PINN is applied to these data to extract inhomogeneous wave velocity data, which can indicate mechanical material property variations with respect to length.
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Huang, Hai, Pengcheng Guo, Jianguo Yan, Bo Zhang, and Zhenkai Mao. "Impact of uncertainty in the physics-informed neural network on pressure prediction for water hammer in pressurized pipelines." Journal of Physics: Conference Series 2707, no. 1 (February 1, 2024): 012095. http://dx.doi.org/10.1088/1742-6596/2707/1/012095.
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Abstract In pressurized pipeline systems, accurate prediction of water hammer pressure is crucial for ensuring safe system operation. When the boundary conditions are unknown and measured data is sparse, both traditional methods fully based on physical equations and data-driven neural network methods have difficulty in accurately predicting water hammer pressure. The physics-informed neural network (PINN) overcomes these challenges by simultaneously incorporating measured data and physical equations during the network training process. However, PINN has uncertainties and their impact on the accuracy of pressure prediction is not yet clear. In this study, the valve closing water hammer in a reservoir-pipeline-valve system is taken as the research object, we investigate the influence of the uncertainty of physics and data in the PINN on prediction accuracy by using water hammer equations with various friction models and training data with various noise levels. The results show that using the water hammer equation with the Brunone model, the PINN model has higher prediction accuracy. Furthermore, data noise levels less than 10% have a relatively small impact on pressure prediction accuracy, indicating that the PINN model has good robustness in terms of data noise levels.
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Paris, Peter J. "Fortress Introduction to Black Church History. Ann H. Pinn , Anthony B. Pinn." Journal of Religion 83, no. 3 (July 2003): 449–50. http://dx.doi.org/10.1086/491355.
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Zhou, Meijun, Gang Mei, and Nengxiong Xu. "Enhancing Computational Accuracy in Surrogate Modeling for Elastic–Plastic Problems by Coupling S-FEM and Physics-Informed Deep Learning." Mathematics 11, no. 9 (April 24, 2023): 2016. http://dx.doi.org/10.3390/math11092016.
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Physics-informed neural networks (PINNs) provide a new approach to solving partial differential equations (PDEs), while the properties of coupled physical laws present potential in surrogate modeling. However, the accuracy of PINNs in solving forward problems needs to be enhanced, and solving inverse problems relies on data samples. The smoothed finite element method (S-FEM) can obtain high-fidelity numerical solutions, which are easy to solve for the forward problems of PDEs, but difficult to solve for the inverse problems. To the best of the authors’ knowledge, there has been no prior research on coupling S-FEM and PINN. In this paper, a novel approach that couples S-FEM and PINN is proposed. The proposed approach utilizes S-FEM to synthesize high-fidelity datasets required for PINN inversion, while also improving the accuracy of data-independent PINN in solving forward problems. The proposed approach is applied to solve linear elastic and elastoplastic forward and inverse problems. The computational results demonstrate that the coupling of the S-FEM and PINN exhibits high precision and convergence when solving inverse problems, achieving a maximum relative error of 0.2% in linear elasticity and 5.69% in elastoplastic inversion by using approximately 10,000 data points. The coupling approach also enhances the accuracy of solving forward problems, reducing relative errors by approximately 2–10 times. The proposed coupling of the S-FEM and PINN offers a novel surrogate modeling approach that incorporates knowledge and data-driven techniques, enabling it to solve both forward and inverse problems associated with PDEs with high levels of accuracy and convergence.
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Oldenburg, Jan, Wiebke Wollenberg, Finja Borowski, Klaus-Peter Schmitz, Michael Stiehm, and Alper Öner. "Augmentation of experimentally obtained flow fields by means of Physics Informed Neural Networks (PINN) demonstrated on aneurysm flow." Current Directions in Biomedical Engineering 9, no. 1 (September 1, 2023): 519–23. http://dx.doi.org/10.1515/cdbme-2023-1130.
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Abstract Biofluid mechanics play an important role in the study of the mechanism of cardiovascular diseases and in the development of new implants. For the assessment of hydrodynamic parameters, experimental methods as well as in-silico approaches can be used, such as particle image velocimetry (PIV) and Deep Learning, respectively. Challenges for PIV are the optical access to the region of interest, and time consumption for measuring and post-processing analysis in particular for three dimensional flow. To overcome these limitations state-of-the-art deep learning algorithms could be utilized to augment spatially coarse resolved flow fields. In this study, we demonstrate the use of Physics Informed Neural Networks (PINN) to augment PIV measurement data. To demonstrate a combined workflow, we investigate the flow of a Newtonian fluid through a simplified aneurysm under laminar conditions. Generation of synthetic PIV particle images of a single measurement plane and the corresponding PIV vector calculations were performed as the basis for the PINN algorithm. Based on the Navier-Stokes equations the PINN reconstructs the entire 3D flow field and pressure distribution inside the aneurysm. We observed qualitative agreements between ground through data and PINN predictions. Nevertheless, there are substantial differences in the quantitative, locally resolved comparison of the flow metrics, despite the generally tendency for the PINN algorithm to correctly augment the flow field.
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Liu, Zhixiang, Yuanji Chen, Ge Song, Wei Song, and Jingxiang Xu. "Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics." Mathematics 11, no. 19 (October 1, 2023): 4147. http://dx.doi.org/10.3390/math11194147.
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Physics-Informed Neural Networks (PINNs) improve the efficiency of data utilization by combining physical principles with neural network algorithms and thus ensure that their predictions are consistent and stable with the physical laws. PINNs open up a new approach to address inverse problems in fluid mechanics. Based on the single-relaxation-time lattice Boltzmann method (SRT-LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator, the PINN-SRT-LBM model is proposed in this paper for solving the inverse problem in fluid mechanics. The PINN-SRT-LBM model consists of three components. The first component involves a deep neural network that predicts equilibrium control equations in different discrete velocity directions within the SRT-LBM. The second component employs another deep neural network to predict non-equilibrium control equations, enabling the inference of the fluid’s non-equilibrium characteristics. The third component, a physics-informed function, translates the outputs of the first two networks into physical information. By minimizing the residuals of the physical partial differential equations (PDEs), the physics-informed function infers relevant macroscopic quantities of the flow. The model evolves two sub-models that are applicable to different dimensions, named the PINN-SRT-LBM-I and PINN-SRT-LBM-II models according to the construction of the physics-informed function. The innovation of this work is the introduction of SRT-LBM and discrete velocity models as physical drivers into a neural network through the interpretation function. Therefore, the PINN-SRT-LBM allows a given neural network to handle inverse problems of various dimensions and focus on problem-specific solving. Our experimental results confirm the accurate prediction by this model of flow information at different Reynolds numbers within the computational domain. Relying on the PINN-SRT-LBM models, inverse problems in fluid mechanics can be solved efficiently.
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Souleymane, Zio, Bernard Lamien, Mohamed Beidari, and Tougri Inoussa. "Numerical simulation of pollutants transport in groundwater using deep neural networks informed by physics." Gulf Journal of Mathematics 16, no. 2 (April 8, 2024): 337–52. http://dx.doi.org/10.56947/gjom.v16i2.1863.
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Real-time monitoring of groundwater pollutants is very challenging to implement due to the difficulty and high economic cost. Groundwater numerical modeling is generally used as an alternative to create protection systems and illustrate their effectiveness. Several groundwater simulation software programs are available, however, they are based on numerical methods that require high spatial and temporal meshing, which considerably increases their computational costs and limits their use for optimization process.To solve this problem, a methodology that combines machine learning techniques based on neural networks, flow, and transport equations in underground reservoirs is proposed. This method is known as physics-informed deep neural networks, abbreviated as PINN. In this work, we will show the performance of PINN in predicting the flow and transport of pollutants in an underground reservoir. To evaluate the effectiveness of the PINN, the finite element method with a fine mesh grid is used as a reference solution to validate the PINN results.The results show that PINNS provide very good result in predicting pollutant transport in the reservoir with a relatively low cost.
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Almqvist, Andreas. "Fundamentals of Physics-Informed Neural Networks Applied to Solve the Reynolds Boundary Value Problem." Lubricants 9, no. 8 (August 19, 2021): 82. http://dx.doi.org/10.3390/lubricants9080082.
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This paper presents a complete derivation and design of a physics-informed neural network (PINN) applicable to solve initial and boundary value problems described by linear ordinary differential equations. The objective with this technical note is not to develop a numerical solution procedure which is more accurate and efficient than standard finite element- or finite difference-based methods, but to give a fully explicit mathematical description of a PINN and to present an application example in the context of hydrodynamic lubrication. It is, however, worth noticing that the PINN developed herein, contrary to FEM and FDM, is a meshless method and that training does not require big data which is typical in machine learning.
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Huang, Yi, Zhiyu Zhang, and Xing Zhang. "A Direct-Forcing Immersed Boundary Method for Incompressible Flows Based on Physics-Informed Neural Network." Fluids 7, no. 2 (January 25, 2022): 56. http://dx.doi.org/10.3390/fluids7020056.
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The application of physics-informed neural networks (PINNs) to computational fluid dynamics simulations has recently attracted tremendous attention. In the simulations of PINNs, the collocation points are required to conform to the fluid–solid interface on which no-slip boundary condition is enforced. Here, a novel PINN that incorporates the direct-forcing immersed boundary (IB) method is developed. In the proposed IB-PINN, the boundary conforming requirement in arranging the collocation points is eliminated. Instead, velocity penalties at some marker points are added to the loss function to enforce no-slip condition at the fluid–solid interface. In addition, force penalties at some collocation points are also added to the loss function to ensure compact distribution of the volume force. The effectiveness of IB-PINN in solving incompressible Navier–Stokes equations is demonstrated through the simulation of laminar flow past a circular cylinder that is placed in a channel. The solution obtained using the IB-PINN is compared with two reference solutions obtained using a conventional mesh-based IB method and an ordinary body-fitted grid method. The comparison indicates that the three solutions are in excellent agreement with each other. The influences of some parameters, such as weights for different loss components, numbers of collocation and marker points, hyperparameters in the neural network, etc., on the performance of IB-PINN are also studied. In addition, a transfer learning experiment is conducted on solving Navier–Stokes equations with different Reynolds numbers.
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Gao, Yunpeng, Li Qian, Tianzhi Yao, Zuguo Mo, Jianhai Zhang, Ru Zhang, Enlong Liu, and Yonghong Li. "An Improved Physics-Informed Neural Network Algorithm for Predicting the Phreatic Line of Seepage." Advances in Civil Engineering 2023 (April 15, 2023): 1–11. http://dx.doi.org/10.1155/2023/5499645.
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As new ways to solve partial differential equations (PDEs), physics-informed neural network (PINN) algorithms have received widespread attention and have been applied in many fields of study. However, the standard PINN framework lacks sufficient seepage head data, and the method is difficult to apply effectively in seepage analysis with complex boundary conditions. In addition, the differential type Neumann boundary makes the solution more difficult. This study proposed an improved prediction method based on a PINN with the aim of calculating PDEs with complex boundary conditions such as Neumann boundary conditions, in which the spatial distribution characteristic information is increased by a small amount of measured data and the loss equation is dynamically adjusted by loss weighting coefficients. The measured data are converted into a quadratic regular term and added to the loss function as feature data to guide the update process for the weight and bias coefficient of each neuron in the neural network. A typical geotechnical problem concerning seepage phreatic line determination in a rectangular dam is analyzed to demonstrate the efficiency of the improved method. Compared with the standard PINN algorithm, due to the addition of measurement data and dynamic loss weighting coefficients, the improved PINN algorithm has better convergence and can handle more complex boundary conditions. The results show that the improved method makes it convenient to predict the phreatic line in seepage analysis for geotechnical engineering projects with measured data.
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Xiao, Chaohao, Xiaoqian Zhu, Fukang Yin, and Xiaoqun Cao. "Physics-informed neural network for solving coupled Korteweg-de Vries equations." Journal of Physics: Conference Series 2031, no. 1 (September 1, 2021): 012056. http://dx.doi.org/10.1088/1742-6596/2031/1/012056.
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Abstract The studies of coupled partial differential equations are focus of engineering and applied mathematics. Although traditional numerical methods have been widely used, researchers are still looking for new methods to solve coupled partial differential equations. In this paper, physical information neural network (PINN) is introduced to solve one-dimensional coupled Korteweg-de Vries (cKdV) equations. Compared with the traditional neural network, the innovation of PINN is to embed the physical constraints of the equations into the network loss function. Moreover, within the acceptable relative error range, the solution can take a longer single time step than the presently available. The results revealed that PINN can solve the cKdV equations with reasonable errors only by training a small amount of data.
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Park, Yongsung, Seunghyun Yoon, Peter Gerstoft, and Woojae Seong. "Physics-informed neural network-based predictions of ocean acoustic pressure fields." Journal of the Acoustical Society of America 155, no. 3_Supplement (March 1, 2024): A44. http://dx.doi.org/10.1121/10.0026740.
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Physics-informed neural network (PINN) trains the network using sampled data and encodes the underlying physical laws governing the dataset, such as partial differential equations (PDEs). A trained PINN can predict data at locations beyond the sampled data positions. The ocean acoustic pressure field satisfies PDEs, Helmholtz equations. We present a method utilizing PINN for predicting the underwater acoustic pressure field. Our approach trains the network by fitting sampled data, embedding PDEs, and enforcing pressure-release surface boundary conditions. We demonstrate our approach under various scenarios. By incorporating PDE information into a neural network, our method captures more accurate solutions than purely data-driven methods. This approach helps enhance the information content of sampled data when dealing with a limited amount of data.
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Psaros, Apostolos F., Kenji Kawaguchi, and George Em Karniadakis. "Meta-learning PINN loss functions." Journal of Computational Physics 458 (June 2022): 111121. http://dx.doi.org/10.1016/j.jcp.2022.111121.
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Conchubhair, Brian Ó., and Greagóir Ó. Dúill. "Fearann Pinn: Filíocht 1900-1999." Comhar 60, no. 12 (2000): 25. http://dx.doi.org/10.2307/25574160.
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Zhang, Jianying. "Physics-Informed Neural Networks for Bingham Fluid Flow Simulation Coupled with an Augmented Lagrange Method." AppliedMath 3, no. 3 (June 30, 2023): 525–51. http://dx.doi.org/10.3390/appliedmath3030028.
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As a class of non-Newtonian fluids with yield stresses, Bingham fluids possess both solid and liquid phases separated by implicitly defined non-physical yield surfaces, which makes the standard numerical discretization challenging. The variational reformulation established by Duvaut and Lions, coupled with an augmented Lagrange method (ALM), brings about a finite element approach, whereas the inevitable local mesh refinement and preconditioning of the resulting large-scaled ill-conditioned linear system can be involved. Inspired by the mesh-free feature and architecture flexibility of physics-informed neural networks (PINNs), an ALM-PINN approach to steady-state Bingham fluid flow simulation, with dynamically adaptable weights, is developed and analyzed in this work. The PINN setting enables not only a pointwise ALM formulation but also the learning of families of (physical) parameter-dependent numerical solutions through one training process, and the incorporation of ALM into a PINN induces a more feasible loss function for deep learning. Numerical results obtained via the ALM-PINN training on one- and two-dimensional benchmark models are presented to validate the proposed scheme. The efficacy and limitations of the relevant loss formulation and optimization algorithms are also discussed to motivate some directions for future research.
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Zhai, Hanfeng, and Timothy Sands. "Comparison of Deep Learning and Deterministic Algorithms for Control Modeling." Sensors 22, no. 17 (August 24, 2022): 6362. http://dx.doi.org/10.3390/s22176362.
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Controlling nonlinear dynamics arises in various engineering fields. We present efforts to model the forced van der Pol system control using physics-informed neural networks (PINN) compared to benchmark methods, including idealized nonlinear feedforward (FF) control, linearized feedback control (FB), and feedforward-plus-feedback combined (C). The aim is to implement circular trajectories in the state space of the van der Pol system. A designed benchmark problem is used for testing the behavioral differences of the disparate controllers and then investigating controlled schemes and systems of various extents of nonlinearities. All methods exhibit a short initialization accompanying arbitrary initialization points. The feedforward control successfully converges to the desired trajectory, and PINN executes good controls with higher stochasticity observed for higher-order terms based on the phase portraits. In contrast, linearized feedback control and combined feed-forward plus feedback failed. Varying trajectory amplitudes revealed that feed-forward, linearized feedback control, and combined feed-forward plus feedback control all fail for unity nonlinear damping gain. Traditional control methods display a robust fluctuation for higher-order terms. For some various nonlinearities, PINN failed to implement the desired trajectory instead of becoming “trapped” in the phase of small radius, yet idealized nonlinear feedforward successfully implemented controls. PINN generally exhibits lower relative errors for varying targeted trajectories. However, PINN also shows evidently higher computational burden compared with traditional control theory methods, with at least more than 30 times longer control time compared with benchmark idealized nonlinear feed-forward control. This manuscript proposes a comprehensive comparative study for future controller employment considering deterministic and machine learning approaches.
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Kim, Jungeun, Kookjin Lee, Dongeun Lee, Sheo Yon Jhin, and Noseong Park. "DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 9 (May 18, 2021): 8146–54. http://dx.doi.org/10.1609/aaai.v35i9.16992.
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We present a method for learning dynamics of complex physical processes described by time-dependent nonlinear partial differential equations (PDEs). Our particular interest lies in extrapolating solutions in time beyond the range of temporal domain used in training. Our choice for a baseline method is physics-informed neural network (PINN) because the method parameterizes not only the solutions, but also the equations that describe the dynamics of physical processes. We demonstrate that PINN performs poorly on extrapolation tasks in many benchmark problems. To address this, we propose a novel method for better training PINN and demonstrate that our newly enhanced PINNs can accurately extrapolate solutions in time. Our method shows up to 72% smaller errors than state-of-the-art methods in terms of the standard L2-norm metric.
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Ma, Yaoyao, Xiaoyu Xu, Shuai Yan, and Zhuoxiang Ren. "A Preliminary Study on the Resolution of Electro-Thermal Multi-Physics Coupling Problem Using Physics-Informed Neural Network (PINN)." Algorithms 15, no. 2 (February 1, 2022): 53. http://dx.doi.org/10.3390/a15020053.
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The problem of electro-thermal coupling is widely present in the integrated circuit (IC). The accuracy and efficiency of traditional solution methods, such as the finite element method (FEM), are tightly related to the quality and density of mesh construction. Recently, PINN (physics-informed neural network) was proposed as a method for solving differential equations. This method is mesh free and generalizes the process of solving PDEs regardless of the equations’ structure. Therefore, an experiment is conducted to explore the feasibility of PINN in solving electro-thermal coupling problems, which include the electrokinetic field and steady-state thermal field. We utilize two neural networks in the form of sequential training to approximate the electric field and the thermal field, respectively. The experimental results show that PINN provides good accuracy in solving electro-thermal coupling problems.
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Xue, Chenyu, Bo Jiang, Jiangong Zhu, Xuezhe Wei, and Haifeng Dai. "An Enhanced Single-Particle Model Using a Physics-Informed Neural Network Considering Electrolyte Dynamics for Lithium-Ion Batteries." Batteries 9, no. 10 (October 15, 2023): 511. http://dx.doi.org/10.3390/batteries9100511.
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As power sources for electric vehicles, lithium-ion batteries (LIBs) have many advantages, such as high energy density and wide temperature range. In the algorithm design process for LIBs, various battery models with different model structures are needed, among which the electrochemical model is widely used due to its high accuracy. However, the electrochemical model is composed of multiple nonlinear partial differential equations (PDEs) that make the simulating process time-consuming. In this paper, a physics-informed neural network single-particle model (PINN SPM) is proposed to improve the accuracy of the single-particle model (SPM) under high C-rates, while ensuring high solving speed. In PINN SPM, an SPM-Net is designed to solve the distribution of lithium-ion concentration in the electrolyte. In the neural network learning process, a loss function is designed based on the physical constraints brought by the PDEs, which reduces the error of the neural network under dynamic working conditions. Finally, the PINN SPM proposed in this paper can achieve a maximum relative error of up to 1.2% compared with the high-fidelity data generated from the P2D model under various conditions. Additionally, the PINN SPM is 20.8% faster than traditional numerical solution methods with the same computational resources.
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Cloud, Nathan, Benjamin M. Goldsberry, and Michael R. Haberman. "Accurate and computationally efficient basis function generation using physics informed neural networks." Journal of the Acoustical Society of America 155, no. 3_Supplement (March 1, 2024): A143. http://dx.doi.org/10.1121/10.0027098.
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Basis functions that can accurately represent simulated or measured acoustic pressure fields with a small number of degrees of freedom is of great use across various applications, including finite element methods, model order reduction, and compressive sensing. In a previous work [B. M. Goldsberry, J. Acoust. Soc. Am. 153, A193 (2023)], basis functions were derived for an element in a given mesh using a combination of interpolation functions defined on the boundaries of the element and the Helmholtz-Kirchhoff (HK) integral. This forms a new interpolatory basis set that efficiently and accurately represents the interior of the element. However, the previous analysis was limited to a two-dimensional rectangular element. In this work, physics informed neural networks (PINN) are investigated as a means to generate HK basis functions for general element shapes. PINNs have been previously shown to accurately learn solutions to parameterized partial differential equations. The element geometry parameterization and the boundary interpolation functions are given as inputs to the PINN, and the output of the PINN consists of the physically accurate basis functions within the element. Details on the implementation and training requirements on the PINN to achieve a desired accuracy will be discussed. [Work supported by ONR.]
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Voigt, Jorrit, and Michael Moeckel. "Modelling dynamic 3D heat transfer in laser material processing based on physics informed neural networks." EPJ Web of Conferences 266 (2022): 02010. http://dx.doi.org/10.1051/epjconf/202226602010.
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Machine learning algorithms make predictions by fitting highly parameterized nonlinear functions to massive amounts of data. Yet those models are not necessarily consistent with physical laws and offer limited interpretability. Extending machine learning models by introducing scientific knowledge in the optimization problem is known as physics-based and data-driven modelling. A promising development are physics informed neural networks (PINN) which ensure consistency to both physical laws and measured data. The aim of this research is to model the time-dependent temperature profile in bulk materials following the passage of a moving laser focus by a PINN. The results from the PINN agree essentially with finite element simulations, proving the suitability of the approach. New perspectives for applications in laser material processing arise when PINNs are integrated in monitoring systems or used for model predictive control.
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Yokota, Kazuya, Takahiko Kurahashi, and Masajiro Abe. "Physics-informed neural network for acoustic resonance analysis in a one-dimensional acoustic tube." Journal of the Acoustical Society of America 156, no. 1 (July 1, 2024): 30–43. http://dx.doi.org/10.1121/10.0026459.
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This study devised a physics-informed neural network (PINN) framework to solve the wave equation for acoustic resonance analysis. The proposed analytical model, ResoNet, minimizes the loss function for periodic solutions and conventional PINN loss functions, thereby effectively using the function approximation capability of neural networks while performing resonance analysis. Additionally, it can be easily applied to inverse problems. The resonance in a one-dimensional acoustic tube, and the effectiveness of the proposed method was validated through the forward and inverse analyses of the wave equation with energy-loss terms. In the forward analysis, the applicability of PINN to the resonance problem was evaluated via comparison with the finite-difference method. The inverse analysis, which included identifying the energy loss term in the wave equation and design optimization of the acoustic tube, was performed with good accuracy.
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Cheng, Chen, and Guang-Tao Zhang. "Deep Learning Method Based on Physics Informed Neural Network with Resnet Block for Solving Fluid Flow Problems." Water 13, no. 4 (February 5, 2021): 423. http://dx.doi.org/10.3390/w13040423.
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Solving fluid dynamics problems mainly rely on experimental methods and numerical simulation. However, in experimental methods it is difficult to simulate the physical problems in reality, and there is also a high-cost to the economy while numerical simulation methods are sensitive about meshing a complicated structure. It is also time-consuming due to the billion degrees of freedom in relevant spatial-temporal flow fields. Therefore, constructing a cost-effective model to settle fluid dynamics problems is of significant meaning. Deep learning (DL) has great abilities to handle strong nonlinearity and high dimensionality that attracts much attention for solving fluid problems. Unfortunately, the proposed surrogate models in DL are almost black-box models and lack interpretation. In this paper, the Physical Informed Neural Network (PINN) combined with Resnet blocks is proposed to solve fluid flows depending on the partial differential equations (i.e., Navier-Stokes equation) which are embedded into the loss function of the deep neural network to drive the model. In addition, the initial conditions and boundary conditions are also considered in the loss function. To validate the performance of the PINN with Resnet blocks, Burger’s equation with a discontinuous solution and Navier-Stokes (N-S) equation with continuous solution are selected. The results show that the PINN with Resnet blocks (Res-PINN) has stronger predictive ability than traditional deep learning methods. In addition, the Res-PINN can predict the whole velocity fields and pressure fields in spatial-temporal fluid flows, the magnitude of the mean square error of the fluid flow reaches to 10−5. The inverse problems of the fluid flows are also well conducted. The errors of the inverse parameters are 0.98% and 3.1% in clean data and 0.99% and 3.1% in noisy data.
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Du, Meiyuan, Chi Zhang, Sheng Xie, Fang Pu, Da Zhang, and Deyu Li. "Investigation on aortic hemodynamics based on physics-informed neural network." Mathematical Biosciences and Engineering 20, no. 7 (2023): 11545–67. http://dx.doi.org/10.3934/mbe.2023512.
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<abstract> <p>Pressure in arteries is difficult to measure non-invasively. Although computational fluid dynamics (CFD) provides high-precision numerical solutions according to the basic physical equations of fluid mechanics, it relies on precise boundary conditions and complex preprocessing, which limits its real-time application. Machine learning algorithms have wide applications in hemodynamic research due to their powerful learning ability and fast calculation speed. Therefore, we proposed a novel method for pressure estimation based on physics-informed neural network (PINN). An ideal aortic arch model was established according to the geometric parameters from human aorta, and we performed CFD simulation with two-way fluid-solid coupling. The simulation results, including the space-time coordinates, the velocity and pressure field, were obtained as the dataset for the training and validation of PINN. Nondimensional Navier-Stokes equations and continuity equation were employed for the loss function of PINN, to calculate the velocity and relative pressure field. Post-processing was proposed to fit the absolute pressure of the aorta according to the linear relationship between relative pressure, elastic modulus and displacement of the vessel wall. Additionally, we explored the sensitivity of the PINN to the vascular elasticity, blood viscosity and blood velocity. The velocity and pressure field predicted by PINN yielded good consistency with the simulated values. In the interested region of the aorta, the relative errors of maximum and average absolute pressure were 7.33% and 5.71%, respectively. The relative pressure field was found most sensitive to blood velocity, followed by blood viscosity and vascular elasticity. This study has proposed a method for intra-vascular pressure estimation, which has potential significance in the diagnosis of cardiovascular diseases.</p> </abstract>
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Hariri, I., A. Radid, and K. Rhofir. "Physics-informed neural networks for the reaction-diffusion Brusselator model." Mathematical Modeling and Computing 11, no. 2 (2024): 448–54. http://dx.doi.org/10.23939/mmc2024.02.448.
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In this work, we are interesting in solving the 1D and 2D nonlinear stiff reaction-diffusion Brusselator system using a machine learning technique called Physics-Informed Neural Networks (PINNs). PINN has been successful in a variety of science and engineering disciplines due to its ability of encoding physical laws, given by the PDE, into the neural network loss function in a way where the network must not only conform to the measurements, initial and boundary conditions, but also satisfy the governing equations. The utilization of PINN for Brusselator system is still in its infancy, with many questions to resolve. Performance of the framework is tested by solving some one and two dimensional problems with comparable numerical or analytical results. Validation of the results is investigated in terms of absolute error. The results showed that our PINN has well performed by producing a good accuracy on the given problems.
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Deng, Yawen, Changchang Chen, Qingxin Wang, Xiaohe Li, Zide Fan, and Yunzi Li. "Modeling a Typical Non-Uniform Deformation of Materials Using Physics-Informed Deep Learning: Applications to Forward and Inverse Problems." Applied Sciences 13, no. 7 (April 3, 2023): 4539. http://dx.doi.org/10.3390/app13074539.
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Numerical methods, such as finite element or finite difference, have been widely used in the past decades for modeling solid mechanics problems by solving partial differential equations (PDEs). Differently from the traditional computational paradigm employed in numerical methods, physics-informed deep learning approximates the physics domains using a neural network and embeds physics laws to regularize the network. In this work, a physics-informed neural network (PINN) is extended for application to linear elasticity problems that arise in modeling non-uniform deformation for a typical open-holed plate specimen. The main focus will be on investigating the performance of a conventional PINN approach to modeling non-uniform deformation with high stress concentration in relation to solid mechanics involving forward and inverse problems. Compared to the conventional finite element method, our results show the promise of using PINN in modeling the non-uniform deformation of materials with the occurrence of both forward and inverse problems.
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Tan, Chenkai, Yingfeng Cai, Hai Wang, Xiaoqiang Sun, and Long Chen. "Vehicle State Estimation Combining Physics-Informed Neural Network and Unscented Kalman Filtering on Manifolds." Sensors 23, no. 15 (July 25, 2023): 6665. http://dx.doi.org/10.3390/s23156665.
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This paper proposes a novel vehicle state estimation (VSE) method that combines a physics-informed neural network (PINN) and an unscented Kalman filter on manifolds (UKF-M). This VSE aimed to achieve inertial measurement unit (IMU) calibration and provide comprehensive information on the vehicle’s dynamic state. The proposed method leverages a PINN to eliminate IMU drift by constraining the loss function with ordinary differential equations (ODEs). Then, the UKF-M is used to estimate the 3D attitude, velocity, and position of the vehicle more accurately using a six-degrees-of-freedom vehicle model. Experimental results demonstrate that the proposed PINN method can learn from multiple sensors and reduce the impact of sensor biases by constraining the ODEs without affecting the sensor characteristics. Compared to the UKF-M algorithm alone, our VSE can better estimate vehicle states. The proposed method has the potential to automatically reduce the impact of sensor drift during vehicle operation, making it more suitable for real-world applications.
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Zhang, Runlin, Nuo Xu, Kai Zhang, Lei Wang, and Gui Lu. "A Parametric Physics-Informed Deep Learning Method for Probabilistic Design of Thermal Protection Systems." Energies 16, no. 9 (April 29, 2023): 3820. http://dx.doi.org/10.3390/en16093820.
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Precise and efficient calculations are necessary to accurately assess the effects of thermal protection system (TPS) uncertainties on aerospacecrafts. This paper presents a probabilistic design methodology for TPSs based on physics-informed neural networks (PINNs) with parametric uncertainty. A typical thermal coating system is used to investigate the impact of uncertainty on the thermal properties of insulation materials and to evaluate the resulting temperature distribution. A sensitivity analysis is conducted to identify the influence of the parameters on the thermal response. The results show that PINNs can produce quick and accurate predictions of the temperature of insulation materials. The accuracy of the PINN model is comparable to that of a response surface surrogate model. Still, the computational time required by the PINN model is only a fraction of the latter. Considering both computational efficiency and accuracy, the PINN model can be used as a high-precision surrogate model to guide the TPS design effectively.
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Zhong, Linlin, Bingyu Wu, and Yifan Wang. "Accelerating Physics-Informed Neural Network based 1-D arc simulation by meta learning." Journal of Physics D: Applied Physics, January 25, 2023. http://dx.doi.org/10.1088/1361-6463/acb604.
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Abstract Physics-Informed Neural Networks (PINNs) have a wide range of applications as an alternative to traditional numerical methods in plasma simulation. However, in some specific cases of PINN-based modeling, a well-trained PINN may require tens of thousands of optimizing iterations during training stage for complex modeling and huge neural networks, which is sometimes very time-consuming. In this work, we propose a meta-learning method, namely Meta-PINN, to reduce the training time of PINN-based 1-D arc simulation. In Meta-PINN, the meta network is first trained by a two-loop optimization on various training tasks of plasma modeling, and then used to initialize the PINN-based network for new tasks. We demonstrate the power of Meta-PINN by four cases corresponding to 1-D arc models at different boundary temperatures, arc radii, arc pressures, and gas mixtures. We found that a well-trained meta network can produce good initial weights for PINN-based arc models even at conditions slightly outside of training range. The speed-up in terms of relative L2 error by Meta-PINN ranges from 1.1x to 6.9x in the cases we studied. The results indicate that Meta-PINN is an effective method for accelerating the PINN-based 1-D arc simulation.
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Zhong, Linlin, Bingyu Wu, and Yifan Wang. "Low-temperature plasma simulation based on physics-informed neural networks: frameworks and preliminary applications." Physics of Fluids, July 26, 2022. http://dx.doi.org/10.1063/5.0106506.
Full textAbstract:
Plasma simulation is an important and sometimes only approach to investigating plasma behavior. In this work, we propose two general AI-driven frameworks for low-temperature plasma simulation: Coefficient-Subnet Physics-Informed Neural Network (CS-PINN) and Runge-Kutta Physics-Informed Neural Network (RK-PINN). CS-PINN uses either a neural network or an interpolation function (e.g. spline function) as the subnet to approximate solution-dependent coefficients (e.g. electron-impact cross sections, thermodynamic properties, transport coefficients, et al.) in plasma equations. On the basis of this, RK-PINN incorporates the implicit Runge-Kutta formalism in neural networks to achieve a large-time-step prediction of transient plasmas. Both CS-PINN and RK-PINN learn the complex non-linear relationship mapping from spatio-temporal space to equation's solution. Based on these two frameworks, we demonstrate preliminary applications by four cases covering plasma kinetic and fluid modeling. The results verify that both CS-PINN and RK-PINN have good performance in solving plasma equations. Moreover, RK-PINN has ability of yielding a good solution for transient plasma simulation with not only large time step but also limited noisy sensing data.