Статті в журналах: "Differentiable simulation"

1

Stanziola, Antonio, Simon Arridge, Ben Cox, and Bradley Treeby. "Application of differentiable programming to wave simulation." Journal of the Acoustical Society of America 155, no. 3_Supplement (March 1, 2024): A106. http://dx.doi.org/10.1121/10.0026968.

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Wave simulations play a crucial role in a wide range of scientific and engineering applications, including seismic imaging, optical design, and acoustic modeling. Here we explore the advantages of differentiable programming in the context of acoustic wave simulation. Differentiable programming enables us to treat wave simulation as a differentiable function, allowing for the automatic computation of gradients with respect to any continuous input parameter. We demonstrate how this approach can be applied to various types of wave simulations, such as Pseudo-spectral time-domain solvers or iterative solvers. Implementing wave simulators via differentiable programming achieves several benefits. First, it enables efficient and accurate sensitivity analysis: This is particularly valuable for optimization and uncertainty quantification tasks. Second, it facilitates the incorporation of wave simulations into machine learning frameworks, enabling the integration of simulation-based models with data-driven approaches. Third, differentiable programming can accelerate the calibration and inversion of wave simulation models, making it easier to match simulated results to observed data. We present practical examples and discuss potential applications in fields such as geophysics and medical imaging. Our findings highlight the potential of this approach to advance the state-of-the-art in wave simulation techniques and their integration into larger computational pipelines.

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2

Viswanathan, Venkatasubramanian. "(Invited) Multi-Physics Modeling of Electrochemical Interfacial Phenomena." ECS Meeting Abstracts MA2024-02, no. 26 (November 22, 2024): 2100. https://doi.org/10.1149/ma2024-02262100mtgabs.

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In this talk, I will discuss our simulation approach which is based on creating an end-to-end differentiable simulation stack for electrochemical phenomena and corresponding validation from 1nm - 1mm. DFT simulations, paired with molecular dynamics, chained to phase-field simulations which are then integrated through reduced-order models in pseudo-2D simulations, enabling simulation capability of interfacial phenomena in a fully differentiable way. I will discuss these results in the context of solid-state batteries.

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3

Son, Sanghyun, Yi-Ling Qiao, Jason Sewall, and Ming C. Lin. "Differentiable Hybrid Traffic Simulation." ACM Transactions on Graphics 41, no. 6 (November 30, 2022): 1–10. http://dx.doi.org/10.1145/3550454.3555492.

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We introduce a novel differentiable hybrid traffic simulator , which simulates traffic using a hybrid model of both macroscopic and microscopic models and can be directly integrated into a neural network for traffic control and flow optimization. This is the first differentiable traffic simulator for macroscopic and hybrid models that can compute gradients for traffic states across time steps and inhomogeneous lanes. To compute the gradient flow between two types of traffic models in a hybrid framework, we present a novel intermediate conversion component that bridges the lanes in a differentiable manner as well. We also show that we can use analytical gradients to accelerate the overall process and enhance scalability. Thanks to these gradients, our simulator can provide more efficient and scalable solutions for complex learning and control problems posed in traffic engineering than other existing algorithms. Refer to https://sites.google.com/umd.edu/diff-hybrid-traffic-sim for our project.

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4

Wang, Ying, Jasper Verheul, Sang-Hoon Yeo, Nima Khademi Kalantari, and Shinjiro Sueda. "Differentiable Simulation of Inertial Musculotendons." ACM Transactions on Graphics 41, no. 6 (November 30, 2022): 1–11. http://dx.doi.org/10.1145/3550454.3555490.

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We propose a simple and practical approach for incorporating the effects of muscle inertia, which has been ignored by previous musculoskeletal simulators in both graphics and biomechanics. We approximate the inertia of the muscle by assuming that muscle mass is distributed along the centerline of the muscle. We express the motion of the musculotendons in terms of the motion of the skeletal joints using a chain of Jacobians, so that at the top level, only the reduced degrees of freedom of the skeleton are used to completely drive both bones and musculotendons. Our approach can handle all commonly used musculotendon path types, including those with multiple path points and wrapping surfaces. For muscle paths involving wrapping surfaces, we use neural networks to model the Jacobians, trained using existing wrapping surface libraries, which allows us to effectively handle the Jacobian discontinuities that occur when musculotendon paths collide with wrapping surfaces. We demonstrate support for higher-order time integrators, complex joints, inverse dynamics, Hill-type muscle models, and differentiability. In the limit, as the muscle mass is reduced to zero, our approach gracefully degrades to traditional simulators without support for muscle inertia. Finally, it is possible to mix and match inertial and non-inertial musculotendons, depending on the application.

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5

Schoenholz, Samuel S., and Ekin D. Cubuk. "JAX, M.D. A framework for differentiable physics*." Journal of Statistical Mechanics: Theory and Experiment 2021, no. 12 (December 1, 2021): 124016. http://dx.doi.org/10.1088/1742-5468/ac3ae9.

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Abstract We introduce JAX MD, a software package for performing differentiable physics simulations with a focus on molecular dynamics. JAX MD includes a number of physics simulation environments, as well as interaction potentials and neural networks that can be integrated into these environments without writing any additional code. Since the simulations themselves are differentiable functions, entire trajectories can be differentiated to perform meta-optimization. These features are built on primitive operations, such as spatial partitioning, that allow simulations to scale to hundreds-of-thousands of particles on a single GPU. These primitives are flexible enough that they can be used to scale up workloads outside of molecular dynamics. We present several examples that highlight the features of JAX MD including: integration of graph neural networks into traditional simulations, meta-optimization through minimization of particle packings, and a multi-agent flocking simulation. JAX MD is available at https://www.github.com/google/jax-md.

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6

Le Lidec, Quentin, Igor Kalevatykh, Ivan Laptev, Cordelia Schmid, and Justin Carpentier. "Differentiable Simulation for Physical System Identification." IEEE Robotics and Automation Letters 6, no. 2 (April 2021): 3413–20. http://dx.doi.org/10.1109/lra.2021.3062323.

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7

Li 李, Yin 寅., Chirag Modi, Drew Jamieson, Yucheng 宇澄 Zhang 张, Libin 利彬 Lu 陆, Yu 雨. Feng 冯, François Lanusse, and Leslie Greengard. "Differentiable Cosmological Simulation with the Adjoint Method." Astrophysical Journal Supplement Series 270, no. 2 (February 1, 2024): 36. http://dx.doi.org/10.3847/1538-4365/ad0ce7.

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Abstract Rapid advances in deep learning have brought not only a myriad of powerful neural networks, but also breakthroughs that benefit established scientific research. In particular, automatic differentiation (AD) tools and computational accelerators like GPUs have facilitated forward modeling of the Universe with differentiable simulations. Based on analytic or automatic backpropagation, current differentiable cosmological simulations are limited by memory, and thus are subject to a trade-off between time and space/mass resolution, usually sacrificing both. We present a new approach free of such constraints, using the adjoint method and reverse time integration. It enables larger and more accurate forward modeling at the field level, and will improve gradient-based optimization and inference. We implement it in an open-source particle-mesh (PM) N-body library pmwd (PM with derivatives). Based on the powerful AD system JAX, pmwd is fully differentiable, and is highly performant on GPUs.

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8

Su, Haozhe, Xuan Li, Tao Xue, Chenfanfu Jiang, and Mridul Aanjaneya. "A Generalized Constitutive Model for Versatile MPM Simulation and Inverse Learning with Differentiable Physics." Proceedings of the ACM on Computer Graphics and Interactive Techniques 6, no. 3 (August 16, 2023): 1–20. http://dx.doi.org/10.1145/3606925.

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We present a generalized constitutive model for versatile physics simulation of inviscid fluids, Newtonian viscosity, hyperelasticity, viscoplasticity, elastoplasticity, and other physical effects that arise due to a mixture of these behaviors. The key ideas behind our formulation are the design of a generalized Kirchhoff stress tensor that can describe hyperelasticity, Newtonian viscosity and inviscid fluids, and the use of pre-projection and post-correction rules for simulating material behaviors that involve plasticity, including elastoplasticity and viscoplasticity. We show how our generalized Kirchhoff stress tensor can be coupled together into a generalized constitutive model that allows the simulation of diverse material behaviors by only changing parameter values. We present several side-by-side comparisons with physics simulations for specific constitutive models to show that our generalized model produces visually similar results. More notably, our formulation allows for inverse learning of unknown material properties directly from data using differentiable physics simulations. We present several 3D simulations to highlight the robustness of our method, even with multiple different materials. To the best of our knowledge, our approach is the first to recover the knowledge of unknown material properties without making explicit assumptions about the data.

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9

Stuyck, Tuur, and Hsiao-yu Chen. "DiffXPBD." Proceedings of the ACM on Computer Graphics and Interactive Techniques 6, no. 3 (August 16, 2023): 1–14. http://dx.doi.org/10.1145/3606923.

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We present DiffXPBD, a novel and efficient analytical formulation for the differentiable position-based simulation of compliant constrained dynamics (XPBD). Our proposed method allows computation of gradients of numerous parameters with respect to a goal function simultaneously leveraging a performant simulation model. The method is efficient, thus enabling differentiable simulations of high resolution geometries and degrees of freedom (DoFs). Collisions are naturally included in the framework. Our differentiable model allows a user to easily add additional optimization variables. Every control variable gradient requires the computation of only a few partial derivatives which can be computed using automatic differentiation code. We demonstrate the efficacy of the method with examples such as elastic cloth and volumetric material parameter estimation, initial value optimization, optimizing for underlying body shape and pose by only observing the clothing, and optimizing a time-varying external force sequence to match sparse keyframe shapes at specific times. Our approach demonstrates excellent efficiency and we demonstrate this on high resolution meshes with optimizations involving over 26 million degrees of freedom. Making an existing solver differentiable requires only a few modifications and the model is compatible with both modern CPU and GPU multi-core hardware.

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10

Numerow, Logan, Yue Li, Stelian Coros, and Bernhard Thomaszewski. "Differentiable Voronoi Diagrams for Simulation of Cell-Based Mechanical Systems." ACM Transactions on Graphics 43, no. 4 (July 19, 2024): 1–11. http://dx.doi.org/10.1145/3658152.

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Navigating topological transitions in cellular mechanical systems is a significant challenge for existing simulation methods. While abstract models lack predictive capabilities at the cellular level, explicit network representations struggle with topology changes, and per-cell representations are computationally too demanding for large-scale simulations. To address these challenges, we propose a novel cell-centered approach based on differentiable Voronoi diagrams. Representing each cell with a Voronoi site, our method defines shape and topology of the interface network implicitly. In this way, we substantially reduce the number of problem variables, eliminate the need for explicit contact handling, and ensure continuous geometry changes during topological transitions. Closed-form derivatives of network positions facilitate simulation with Newton-type methods for a wide range of per-cell energies. Finally, we extend our differentiable Voronoi diagrams to enable coupling with arbitrary rigid and deformable boundaries. We apply our approach to a diverse set of examples, highlighting splitting and merging of cells as well as neighborhood changes. We illustrate applications to inverse problems by matching soap foam simulations to real-world images. Comparative analysis with explicit cell models reveals that our method achieves qualitatively comparable results at significantly faster computation times.

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11

Heinrich, Lukas, and Michael Kagan. "Differentiable Matrix Elements with MadJax." Journal of Physics: Conference Series 2438, no. 1 (February 1, 2023): 012137. http://dx.doi.org/10.1088/1742-6596/2438/1/012137.

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Abstract MadJax is a tool for generating and evaluating differentiable matrix elements of high energy scattering processes. As such, it is a step towards a differentiable programming paradigm in high energy physics that facilitates the incorporation of high energy physics domain knowledge, encoded in simulation software, into gradient based learning and optimization pipelines. MadJax comprises two components: (a) a plugin to the general purpose matrix element generator MadGraph that integrates matrix element and phase space sampling code with the JAX differentiable programming framework, and (b) a standalone wrapping code interface for accessing the matrix element code and its gradients, which are computed with automatic differentiation. The MadJax implementation and example applications of simulation based inference and normalizing flow based matrix element modeling, with capabilities enabled uniquely with differentiable matrix elements, are presented.

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12

Du, Tao, Kui Wu, Pingchuan Ma, Sebastien Wah, Andrew Spielberg, Daniela Rus, and Wojciech Matusik. "DiffPD: Differentiable Projective Dynamics." ACM Transactions on Graphics 41, no. 2 (April 30, 2022): 1–21. http://dx.doi.org/10.1145/3490168.

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We present a novel, fast differentiable simulator for soft-body learning and control applications. Existing differentiable soft-body simulators can be classified into two categories based on their time integration methods: Simulators using explicit timestepping schemes require tiny timesteps to avoid numerical instabilities in gradient computation, and simulators using implicit time integration typically compute gradients by employing the adjoint method and solving the expensive linearized dynamics. Inspired by Projective Dynamics ( PD ), we present Differentiable Projective Dynamics ( DiffPD ), an efficient differentiable soft-body simulator based on PD with implicit time integration. The key idea in DiffPD is to speed up backpropagation by exploiting the prefactorized Cholesky decomposition in forward PD simulation. In terms of contact handling, DiffPD supports two types of contacts: a penalty-based model describing contact and friction forces and a complementarity-based model enforcing non-penetration conditions and static friction. We evaluate the performance of DiffPD and observe it is 4–19 times faster compared with the standard Newton’s method in various applications including system identification, inverse design problems, trajectory optimization, and closed-loop control. We also apply DiffPD in a reality-to-simulation ( real-to-sim ) example with contact and collisions and show its capability of reconstructing a digital twin of real-world scenes.

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13

Dorda, D., D. Peter, D. Borer, N. B. Huber, I. Sailer, M. Gross, B. Solenthaler, and B. Thomaszewski. "Differentiable Simulation for Outcome‐Driven Orthognathic Surgery Planning." Computer Graphics Forum 41, no. 8 (December 2022): 53–61. http://dx.doi.org/10.1111/cgf.14623.

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14

Prakash, Amit, and Hardish Kaur. "An efficient hybrid computational technique for solving nonlinear local fractional partial differential equations arising in fractal media." Nonlinear Engineering 7, no. 3 (September 25, 2018): 229–35. http://dx.doi.org/10.1515/nleng-2017-0100.

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AbstractIn present work, nonlinear fractional partial differential equations namely transport equation and Fokker-Planck equation involving local fractional differential operators, are investigated by means of the local fractional homotopy perturbation Sumudu transform method. The proposed method is a coupling of homotopy perturbation method with local fractional Sumudu transform and is used to describe the non-differentiable problems. Numerical simulation results are projected to show the efficiency of the proposed technique.

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15

Michel, Jesse, Kevin Mu, Xuanda Yang, Sai Praveen Bangaru, Elias Rojas Collins, Gilbert Bernstein, Jonathan Ragan-Kelley, Michael Carbin, and Tzu-Mao Li. "Distributions for Compositionally Differentiating Parametric Discontinuities." Proceedings of the ACM on Programming Languages 8, OOPSLA1 (April 29, 2024): 893–922. http://dx.doi.org/10.1145/3649843.

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Computations in physical simulation, computer graphics, and probabilistic inference often require the differentiation of discontinuous processes due to contact, occlusion, and changes at a point in time. Popular differentiable programming languages, such as PyTorch and JAX, ignore discontinuities during differentiation. This is incorrect for parametric discontinuities —conditionals containing at least one real-valued parameter and at least one variable of integration. We introduce Potto, the first differentiable first-order programming language to soundly differentiate parametric discontinuities. We present a denotational semantics for programs and program derivatives and show the two accord. We describe the implementation of Potto, which enables separate compilation of programs. Our prototype implementation overcomes previous compile-time bottlenecks achieving an 88.1x and 441.2x speed up in compile time and a 2.5x and 7.9x speed up in runtime, respectively, on two increasingly large image stylization benchmarks. We showcase Potto by implementing a prototype differentiable renderer with separately compiled shaders.

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16

Bächer, Moritz, Espen Knoop, and Christian Schumacher. "Design and Control of Soft Robots Using Differentiable Simulation." Current Robotics Reports 2, no. 2 (May 10, 2021): 211–21. http://dx.doi.org/10.1007/s43154-021-00052-7.

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17

Du, Tao, Josie Hughes, Sebastien Wah, Wojciech Matusik, and Daniela Rus. "Underwater Soft Robot Modeling and Control With Differentiable Simulation." IEEE Robotics and Automation Letters 6, no. 3 (July 2021): 4994–5001. http://dx.doi.org/10.1109/lra.2021.3070305.

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18

Cea, Alvaro, and Rafael Palacios. "JAX-based aeroelastic simulation engine for differentiable aircraft dynamics." Computer Physics Communications 311 (June 2025): 109547. https://doi.org/10.1016/j.cpc.2025.109547.

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19

Kasim, Muhammad F., Susi Lehtola, and Sam M. Vinko. "DQC: A Python program package for differentiable quantum chemistry." Journal of Chemical Physics 156, no. 8 (February 28, 2022): 084801. http://dx.doi.org/10.1063/5.0076202.

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Automatic differentiation represents a paradigm shift in scientific programming, where evaluating both functions and their derivatives is required for most applications. By removing the need to explicitly derive expressions for gradients, development times can be shortened and calculations can be simplified. For these reasons, automatic differentiation has fueled the rapid growth of a variety of sophisticated machine learning techniques over the past decade, but is now also increasingly showing its value to support ab initio simulations of quantum systems and enhance computational quantum chemistry. Here, we present an open-source differentiable quantum chemistry simulation code and explore applications facilitated by automatic differentiation: (1) calculating molecular perturbation properties, (2) reoptimizing a basis set for hydrocarbons, (3) checking the stability of self-consistent field wave functions, and (4) predicting molecular properties via alchemical perturbations.

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20

Shi, Zhi Guang, and Wei Huo. "Robust Tracking for BLDCM System Based on Unknown Differentiable Deadzone Nonlinearity." Applied Mechanics and Materials 313-314 (March 2013): 530–34. http://dx.doi.org/10.4028/www.scientific.net/amm.313-314.530.

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In this paper, trajectory tracking control of the brushless DC motor system (BLDCMS) with parameter unknown deadzone nonlinearity and viscous friction is investigated. Firstly, a global differential homeomorphism based on the recently established differentiable deadzone model is developed to linearize BLDCMS. Then, a model reference robust controller is presented to suppress the uncertainties. Finally, uniformly ultimate boundedness of the closed-loop system is proved and simulation results show validity of the proposed controller.

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21

Greener, Joe G., and David T. Jones. "Differentiable molecular simulation can learn all the parameters in a coarse-grained force field for proteins." PLOS ONE 16, no. 9 (September 2, 2021): e0256990. http://dx.doi.org/10.1371/journal.pone.0256990.

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Finding optimal parameters for force fields used in molecular simulation is a challenging and time-consuming task, partly due to the difficulty of tuning multiple parameters at once. Automatic differentiation presents a general solution: run a simulation, obtain gradients of a loss function with respect to all the parameters, and use these to improve the force field. This approach takes advantage of the deep learning revolution whilst retaining the interpretability and efficiency of existing force fields. We demonstrate that this is possible by parameterising a simple coarse-grained force field for proteins, based on training simulations of up to 2,000 steps learning to keep the native structure stable. The learned potential matches chemical knowledge and PDB data, can fold and reproduce the dynamics of small proteins, and shows ability in protein design and model scoring applications. Problems in applying differentiable molecular simulation to all-atom models of proteins are discussed along with possible solutions and the variety of available loss functions. The learned potential, simulation scripts and training code are made available at https://github.com/psipred/cgdms.

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22

JUMARIE, GUY. "RIEMANN-CHRISTOFFEL TENSOR IN DIFFERENTIAL GEOMETRY OF FRACTIONAL ORDER APPLICATION TO FRACTAL SPACE-TIME." Fractals 21, no. 01 (March 2013): 1350004. http://dx.doi.org/10.1142/s0218348x13500047.

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By using fractional differences, one recently proposed an alternative to the formulation of fractional differential calculus, of which the main characteristics is a new fractional Taylor series and its companion Rolle's formula which apply to non-differentiable functions. The key is that now we have at hand a differential increment of fractional order which can be manipulated exactly like in the standard Leibniz differential calculus. Briefly the fractional derivative is the quotient of fractional increments. It has been proposed that this calculus can be used to construct a differential geometry on manifold of fractional order. The present paper, on the one hand, refines the framework, and on the other hand, contributes some new results related to arc length of fractional curves, area on fractional differentiable manifold, covariant fractal derivative, Riemann-Christoffel tensor of fractional order, fractional differential equations of fractional geodesic, strip modeling of fractal space time and its relation with Lorentz transformation. The relation with Nottale's fractal space-time theory then appears in quite a natural way.

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23

Kuo, Chao-Hung, Jia-Wei Chen, Yi Yang, Yu-Hao Lan, Shao-Wei Lu, Ching-Fu Wang, Yu-Chun Lo, et al. "A Differentiable Dynamic Model for Musculoskeletal Simulation and Exoskeleton Control." Biosensors 12, no. 5 (May 9, 2022): 312. http://dx.doi.org/10.3390/bios12050312.

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An exoskeleton, a wearable device, was designed based on the user’s physical and cognitive interactions. The control of the exoskeleton uses biomedical signals reflecting the user intention as input, and its algorithm is calculated as an output to make the movement smooth. However, the process of transforming the input of biomedical signals, such as electromyography (EMG), into the output of adjusting the torque and angle of the exoskeleton is limited by a finite time lag and precision of trajectory prediction, which result in a mismatch between the subject and exoskeleton. Here, we propose an EMG-based single-joint exoskeleton system by merging a differentiable continuous system with a dynamic musculoskeletal model. The parameters of each muscle contraction were calculated and applied to the rigid exoskeleton system to predict the precise trajectory. The results revealed accurate torque and angle prediction for the knee exoskeleton and good performance of assistance during movement. Our method outperformed other models regarding the rate of convergence and execution time. In conclusion, a differentiable continuous system merged with a dynamic musculoskeletal model supported the effective and accurate performance of an exoskeleton controlled by EMG signals.

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24

Li, Zhehao, Qingyu Xu, Xiaohan Ye, Bo Ren, and Ligang Liu. "DiffFR: Differentiable SPH-Based Fluid-Rigid Coupling for Rigid Body Control." ACM Transactions on Graphics 42, no. 6 (December 5, 2023): 1–17. http://dx.doi.org/10.1145/3618318.

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Differentiable physics simulation has shown its efficacy in inverse design problems. Given the pervasiveness of the diverse interactions between fluids and solids in life, a differentiable simulator for the inverse design of the motion of rigid objects in two-way fluid-rigid coupling is also demanded. There are two main challenges to develop a differentiable two-way fluid-solid coupling simulator for rigid body control tasks: the ubiquitous, discontinuous contacts in fluid-solid interactions, and the high computational cost of gradient formulation due to the large number of degrees of freedom (DoF) of fluid dynamics. In this work, we propose a novel differentiable SPH-based two-way fluid-rigid coupling simulator to address these challenges. Our purpose is to provide a differentiable simulator for SPH which incorporates a unified representation for both fluids and solids using particles. However, naively differentiating the forward simulation of the particle system encounters gradient explosion issues. We investigate the instability in differentiating the SPH-based fluid-rigid coupling simulator and present a feasible gradient computation scheme to address its differentiability. In addition, we also propose an efficient method to compute the gradient of fluid-rigid coupling without incurring the high computational cost of differentiating the entire high-DoF fluid system. We show the efficacy, scalability, and extensibility of our method in various challenging rigid body control tasks with diverse fluid-rigid interactions and multi-rigid contacts, achieving up to an order of magnitude speedup in optimization compared to baseline methods in experiments.

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Rahman, Jamshaid Ul, Sana Danish, and Dianchen Lu. "Oscillator Simulation with Deep Neural Networks." Mathematics 12, no. 7 (March 23, 2024): 959. http://dx.doi.org/10.3390/math12070959.

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The motivation behind this study is to overcome the complex mathematical formulation and time-consuming nature of traditional numerical methods used in solving differential equations. It seeks an alternative approach for more efficient and simplified solutions. A Deep Neural Network (DNN) is utilized to understand the intricate correlations between the oscillator’s variables and to precisely capture their dynamics by being trained on a dataset of known oscillator behaviors. In this work, we discuss the main challenge of predicting the behavior of oscillators without depending on complex strategies or time-consuming simulations. The present work proposes a favorable modified form of neural structure to improve the strategy for simulating linear and nonlinear harmonic oscillators from mechanical systems by formulating an ANN as a DNN via an appropriate oscillating activation function. The proposed methodology provides the solutions of linear and nonlinear differential equations (DEs) in differentiable form and is a more accurate approximation as compared to the traditional numerical method. The Van der Pol equation with parametric damping and the Mathieu equation are adopted as illustrations. Experimental analysis shows that our proposed scheme outperforms other numerical methods in terms of accuracy and computational cost. We provide a comparative analysis of the outcomes obtained through our proposed approach and those derived from the LSODA algorithm, utilizing numerical techniques, Adams–Bashforth, and the Backward Differentiation Formula (BDF). The results of this research provide insightful information for engineering applications, facilitating improvements in energy efficiency, and scientific innovation.

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Tan, Chuin Wei, Chris J. Pickard, and William C. Witt. "Automatic differentiation for orbital-free density functional theory." Journal of Chemical Physics 158, no. 12 (March 28, 2023): 124801. http://dx.doi.org/10.1063/5.0138429.

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Differentiable programming has facilitated numerous methodological advances in scientific computing. Physics engines supporting automatic differentiation have simpler code, accelerating the development process and reducing the maintenance burden. Furthermore, fully differentiable simulation tools enable direct evaluation of challenging derivatives—including those directly related to properties measurable by experiment—that are conventionally computed with finite difference methods. Here, we investigate automatic differentiation in the context of orbital-free density functional theory (OFDFT) simulations of materials, introducing PROFESS-AD. Its automatic evaluation of properties derived from first derivatives, including functional potentials, forces, and stresses, facilitates the development and testing of new density functionals, while its direct evaluation of properties requiring higher-order derivatives, such as bulk moduli, elastic constants, and force constants, offers more concise implementations than conventional finite difference methods. For these reasons, PROFESS-AD serves as an excellent prototyping tool and provides new opportunities for OFDFT.

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27

Rubio, Gerardo. "The Cauchy-Dirichlet Problem for a Class of Linear Parabolic Differential Equations with Unbounded Coefficients in an Unbounded Domain." International Journal of Stochastic Analysis 2011 (June 22, 2011): 1–35. http://dx.doi.org/10.1155/2011/469806.

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We consider the Cauchy-Dirichlet problem in [0,∞)×D for a class of linear parabolic partial differential equations. We assume that D⊂ℝd is an unbounded, open, connected set with regular boundary. Our hypotheses are unbounded and locally Lipschitz coefficients, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution to the nonhomogeneous Cauchy-Dirichlet problem using stochastic differential equations and parabolic differential equations in bounded domains.

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28

Napolitano, Fabrizio. "Enhancing Spectroscopic Experiment Calibration through Differentiable Programming." Condensed Matter 9, no. 2 (June 5, 2024): 26. http://dx.doi.org/10.3390/condmat9020026.

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In this work, we present an innovative calibration technique leveraging differentiable programming to enhance energy resolution and reduce the energy scale systematic uncertainty in X-ray spectroscopic experiments. This approach is demonstrated using synthetic data and is applicable in general to various spectroscopic measurements. This method extends the scope of differentiable programming for calibration, employing Kernel Density Estimation (KDE) to achieve a target Probability Density Function (PDF) for a fully differentiable model of the calibration. To assess the effectiveness of the calibration, we conduct a toy simulation replicating the entire detector response chain and compare it with a standard calibration. This ensures a robust and reliable calibration methodology, holding promise for improving energy resolution and providing a more versatile and efficient approach without the need for extensive fine-tuning.

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29

de Lima, Amanda, and Daniel Smania. "Central limit theorem for generalized Weierstrass functions." Stochastics and Dynamics 19, no. 01 (January 27, 2019): 1950002. http://dx.doi.org/10.1142/s0219493719500023.

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Let [Formula: see text] be a [Formula: see text] expanding map of the circle and let [Formula: see text] be a [Formula: see text] function. Consider the twisted cohomological equation [Formula: see text] which has a unique bounded solution [Formula: see text]. We show that [Formula: see text] is either [Formula: see text] or continuous but nowhere differentiable. If [Formula: see text] is nowhere differentiable then the Newton quotients of [Formula: see text], after an appropriated normalization, converges in distribution (with respect to the unique absolutely continuous invariant probability of [Formula: see text]) to the normal distribution. In particular, [Formula: see text] is not a Lipschitz continuous function on any subset with positive Lebesgue measure.

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30

Ramos, J. I. "Elements of Differentiable Dynamics and Bifurcation Theory." Applied Mathematical Modelling 14, no. 8 (August 1990): 445. http://dx.doi.org/10.1016/0307-904x(90)90104-d.

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31

Feng, Dapeng, Hylke Beck, Jens de Bruijn, Reetik Kumar Sahu, Yusuke Satoh, Yoshihide Wada, Jiangtao Liu, Ming Pan, Kathryn Lawson та Chaopeng Shen. "Deep dive into hydrologic simulations at global scale: harnessing the power of deep learning and physics-informed differentiable models (δHBV-globe1.0-hydroDL)". Geoscientific Model Development 17, № 18 (26 вересня 2024): 7181–98. http://dx.doi.org/10.5194/gmd-17-7181-2024.

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Abstract. Accurate hydrologic modeling is vital to characterizing how the terrestrial water cycle responds to climate change. Pure deep learning (DL) models have been shown to outperform process-based ones while remaining difficult to interpret. More recently, differentiable physics-informed machine learning models with a physical backbone can systematically integrate physical equations and DL, predicting untrained variables and processes with high performance. However, it is unclear if such models are competitive for global-scale applications with a simple backbone. Therefore, we use – for the first time at this scale – differentiable hydrologic models (full name δHBV-globe1.0-hydroDL, shortened to δHBV here) to simulate the rainfall–runoff processes for 3753 basins around the world. Moreover, we compare the δHBV models to a purely data-driven long short-term memory (LSTM) model to examine their strengths and limitations. Both LSTM and the δHBV models provide competitive daily hydrologic simulation capabilities in global basins, with median Kling–Gupta efficiency values close to or higher than 0.7 (and 0.78 with LSTM for a subset of 1675 basins with long-term discharge records), significantly outperforming traditional models. Moreover, regionalized differentiable models demonstrated stronger spatial generalization ability (median KGE 0.64) than a traditional parameter regionalization approach (median KGE 0.46) and even LSTM for ungauged region tests across continents. Nevertheless, relative to LSTM, the differentiable model was hampered by structural deficiencies for cold or polar regions, highly arid regions, and basins with significant human impacts. This study also sets the benchmark for hydrologic estimates around the world and builds a foundation for improving global hydrologic simulations.

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32

Guo, Yingjia. "The stability of the positive solution for a fractional SIR model." International Journal of Biomathematics 10, no. 01 (November 15, 2016): 1750014. http://dx.doi.org/10.1142/s1793524517500140.

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In order to deal with non-differentiable functions, a modification of the Riemann–Liouville definition is recently proposed which appears to provide a framework for a fractional calculus which is quite parallel with classical calculus. Based on these new results, we study on the fractional SIR model in this paper. First, we give the general solution of the fractional differential equation. And then a unique global positive solution of the SIR model is obtained. Furthermore, we get the Lyapunov stability theory. By using this stability theory, the asymptotic stability of the positive solution is analyzed.

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33

Geminiani, Elena, Giampiero Marra, and Irini Moustaki. "Single- and Multiple-Group Penalized Factor Analysis: A Trust-Region Algorithm Approach with Integrated Automatic Multiple Tuning Parameter Selection." Psychometrika 86, no. 1 (March 2021): 65–95. http://dx.doi.org/10.1007/s11336-021-09751-8.

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AbstractPenalized factor analysis is an efficient technique that produces a factor loading matrix with many zero elements thanks to the introduction of sparsity-inducing penalties within the estimation process. However, sparse solutions and stable model selection procedures are only possible if the employed penalty is non-differentiable, which poses certain theoretical and computational challenges. This article proposes a general penalized likelihood-based estimation approach for single- and multiple-group factor analysis models. The framework builds upon differentiable approximations of non-differentiable penalties, a theoretically founded definition of degrees of freedom, and an algorithm with integrated automatic multiple tuning parameter selection that exploits second-order analytical derivative information. The proposed approach is evaluated in two simulation studies and illustrated using a real data set. All the necessary routines are integrated into the package .

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34

Fu, Zhi-Jun, Wei-Dong Xie, and Xiao-Bin Ning. "Adaptive Nonlinear Tire-Road Friction Force Estimation for Vehicular Systems Based on a Novel Differentiable Friction Model." Mathematical Problems in Engineering 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/201062.

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A novel adaptive nonlinear observer-based parameter estimation scheme using a newly continuously differentiable friction model has been developed to estimate the tire-road friction force. The differentiable friction model is more flexible and suitable for online adaptive identification and control with the advantage of more explicit parameterizable form. Different from conventional estimation methods, the filtered regression estimation parameter is introduced in the novel adaptive laws, which can guarantee that both the observer error and parameter error exponentially converge to zero. Lyapunov theory has been used to prove the stability of the proposed methods. The effectiveness of the estimation algorithm is illustrated via a bus simulation model in the Trucksim software and simulation environment. The relatively accurate tire-road friction force was estimated just by the easily existing sensors signals wheel rotational speed and vehicle speed and the proposed method also displays strong robustness against bounded disturbances.

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35

Raharjo, Jangkung, Adi Soeprijanto, and Hermagasantos Zein. "Multi Dimension of Coarse to Fine Search Method Development for Solving Economic Dispatch." Indonesian Journal of Electrical Engineering and Computer Science 3, no. 1 (June 4, 2016): 1. http://dx.doi.org/10.11591/ijeecs.v3.i1.pp1-9.

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<p>Economic dispatch problem has grown along with the development of electric power business, for example in a competitive electric power business that offers electrical energy in the form of the step function, non-differentiable function. This is not a continuous function so there is no guarantee that those methods can execute the optimization problem well, especially the Lagrange and Direct methods. There are the non-differentiable functions within the optimization will become a challenge that should be solved. This paper proposes Coarse to Fine Search method development to solve the problem. The Coarse to Fine Search is able to work for differentiable or non-differentiable functions, but is only limited maximum three dimensions. The development is done to multi dimension so that it can solve the economic dispatch problem. We named it Multi Dimension of Coarse to Fine Search. The simulation results of eight power plants show the developed method can work well, it is always convergent and fast with the execution time of 2.63 - 38.30 seconds for 25 - 200 population and 50 - 200 delta search.</p>

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36

Burden, Samuel A., S. Shankar Sastry, Daniel E. Koditschek, and Shai Revzen. "Event--Selected Vector Field Discontinuities Yield Piecewise--Differentiable Flows." SIAM Journal on Applied Dynamical Systems 15, no. 2 (January 2016): 1227–67. http://dx.doi.org/10.1137/15m1016588.

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37

Che, Chengfu, Bin Ge, Xiao-Ping Xue, and Qing-Mei Zhou. "W 0 1, P(X) VERSUS C 1 LOCAL MINIMIZERS FOR NONSMOOTH FUNCTIONALS." Mathematical Modelling and Analysis 17, no. 3 (June 1, 2012): 396–402. http://dx.doi.org/10.3846/13926292.2012.686066.

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38

Lima, G. A. B., V. G. Ferreira, E. R. Cirilo, A. Castelo, M. A. C. Candezano, I. V. M. Tasso, D. M. C. Sano, and L. V. A. Scalvi. "A continuously differentiable upwinding scheme for the simulation of fluid flow problems." Applied Mathematics and Computation 218, no. 17 (May 2012): 8614–33. http://dx.doi.org/10.1016/j.amc.2012.02.024.

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39

Vilches, Karina, Eduardo González-Olivares, and Alejandro Rojas-Palma. "Prey herd behavior modeled by a generic non-differentiable functional response." Mathematical Modelling of Natural Phenomena 13, no. 3 (2018): 26. http://dx.doi.org/10.1051/mmnp/2018038.

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Over the past decade, many works have studied an antipredator behavior (APB) named prey herd behavior. Analyzes have been conducted by modifying the classical predator consumption rate to be dependent only on the prey population size assuming the square root functional response. This work focuses analyzing the dynamics of a Gause-type predator-prey model considering that social behavior of prey. However, we model this phenomenon using a Holling type II non-differentiable rational functional response, which is more general than that mentioned above. The studied model exhibits richer dynamics than those with differentiable functional responses, and one the main consequences of including this type of function is the existence of initial values for which the extinction of prey occurs within a finite time for all parameter conditions, which is a direct consequence of the non-uniqueness of the solutions over the vertical axes and of the existence of a separatrix curve dividing the phase plane. A discussion on what represents a well-posed problem from both the mathematical and the ecological points of view is presented. Additionally, the differences in other social behaviors of the prey are also established. Numerical simulations are provided to validate the mathematical results.

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40

Guo, Xinxin, An Guo, and Suping Zhao. "Null-Space-Based Multi-Player Pursuit-Evasion Games Using Minimum and Maximum Approximation Functions." Electronics 11, no. 22 (November 14, 2022): 3729. http://dx.doi.org/10.3390/electronics11223729.

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In this article, pursuit and evasion policies are developed for multi-player pursuit–evasion games, while obstacle avoidance and velocity constraints are considered simultaneously. As minimum and maximum approximation functions are both differentiable, pursuit and evasion objectives can be transformed into solving the corresponding differential expressions. For obstacle avoidance, a modified null-space-based approach is designed, which can ensure that all pursuers and evaders of pursuit–evasions are safe to minimize pursuit objective and maximize evasion objective, respectively. Rigorous theoretical analyses are provided to design constrained pursuit and evasion policies with obstacle avoidance. Finally, the performance of proposed policies is demonstrated by simulation results in 3-dimensional space.

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41

Chen, Xuwen, Xingyu Ni, Bo Zhu, Bin Wang, and Baoquan Chen. "Simulation and optimization of magnetoelastic thin shells." ACM Transactions on Graphics 41, no. 4 (July 2022): 1–18. http://dx.doi.org/10.1145/3528223.3530142.

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Magnetoelastic thin shells exhibit great potential in realizing versatile functionalities through a broad range of combination of material stiffness, remnant magnetization intensity, and external magnetic stimuli. In this paper, we propose a novel computational method for forward simulation and inverse design of magnetoelastic thin shells. Our system consists of two key components of forward simulation and backward optimization. On the simulation side, we have developed a new continuum mechanics model based on the Kirchhoff-Love thin-shell model to characterize the behaviors of a megnetolelastic thin shell under external magnetic stimuli. Based on this model, we proposed an implicit numerical simulator facilitated by the magnetic energy Hessian to treat the elastic and magnetic stresses within a unified framework, which is versatile to incorporation with other thin shell models. On the optimization side, we have devised a new differentiable simulation framework equipped with an efficient adjoint formula to accommodate various PDE-constraint, inverse design problems of magnetoelastic thin-shell structures, in both static and dynamic settings. It also encompasses applications of magnetoelastic soft robots, functional Origami, artworks, and meta-material designs. We demonstrate the efficacy of our framework by designing and simulating a broad array of magnetoelastic thin-shell objects that manifest complicated interactions between magnetic fields, materials, and control policies.

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42

Zhang, Qiongfen, X. H. Tang, and Qi-Ming Zhang. "Existence of Periodic Solutions for a Class of Discrete Hamiltonian Systems." Discrete Dynamics in Nature and Society 2011 (2011): 1–14. http://dx.doi.org/10.1155/2011/463480.

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By applying minimax methods in critical point theory, we prove the existence of periodic solutions for the following discrete Hamiltonian systemsΔ2u(t-1)+∇F(t,u(t))=0, wheret∈ℤ,u∈ℝN,F:ℤ×ℝN→ℝ,F(t,x)is continuously differentiable inxfor everyt∈ℤand isT-periodic int;Tis a positive integer.

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43

Robitzsch, Alexander. "Implementation Aspects in Regularized Structural Equation Models." Algorithms 16, no. 9 (September 18, 2023): 446. http://dx.doi.org/10.3390/a16090446.

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This article reviews several implementation aspects in estimating regularized single-group and multiple-group structural equation models (SEM). It is demonstrated that approximate estimation approaches that rely on a differentiable approximation of non-differentiable penalty functions perform similarly to the coordinate descent optimization approach of regularized SEMs. Furthermore, using a fixed regularization parameter can sometimes be superior to an optimal regularization parameter selected by the Bayesian information criterion when it comes to the estimation of structural parameters. Moreover, the widespread penalty functions of regularized SEM implemented in several R packages were compared with the estimation based on a recently proposed penalty function in the Mplus software. Finally, we also investigate the performance of a clever replacement of the optimization function in regularized SEM with a smoothed differentiable approximation of the Bayesian information criterion proposed by O’Neill and Burke in 2023. The findings were derived through two simulation studies and are intended to guide the practical implementation of regularized SEM in future software pieces.

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44

Wang, Pengyuan, Xinjian Wang, and Yunpeng Wang. "End-to-End Differentiable Physics Temperature Estimation for Permanent Magnet Synchronous Motor." World Electric Vehicle Journal 15, no. 4 (April 21, 2024): 174. http://dx.doi.org/10.3390/wevj15040174.

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Differentiable physics is an approach that effectively combines physical models with deep learning, providing valuable information about physical systems during the training process of neural networks. This integration enhances the generalization ability and ensures better consistency with physical principles. In this work, we propose a framework for estimating the temperature of a permanent magnet synchronous motor by combining neural networks with the differentiable physical thermal model, as well as utilizing the simulation results. In detail, we first implement a differentiable thermal model based on a lumped parameter thermal network within an automatic differentiation framework. Subsequently, we add a neural network to predict thermal resistances, capacitances, and losses in real time and utilize the thermal parameters’ optimized empirical values as the initial output values of the network to improve the accuracy and robustness of the final temperature estimation. We validate the conceivable advantages of the proposed method through extensive experiments based on both synthetic data and real-world data and then provide some further potential applications.

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45

MIN, LEQUAN, and GUANRONG CHEN. "GENERALIZED SYNCHRONIZATION IN AN ARRAY OF NONLINEAR DYNAMIC SYSTEMS WITH APPLICATIONS TO CHAOTIC CNN." International Journal of Bifurcation and Chaos 23, no. 01 (January 2013): 1350016. http://dx.doi.org/10.1142/s0218127413500168.

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This paper establishes some generalized synchronization (GS) theorems for a coupled discrete array of difference systems (CDADS) and a coupled continuous array of differential systems (CCADS). These constructive theorems provide general representations of GS in CDADS and CCADS. Based on these theorems, one can design GS-driven CDADS and CCADS via appropriate (invertible) transformations. As applications, the results are applied to autonomous and nonautonomous coupled Chen cellular neural network (CNN) CDADS and CCADS, discrete bidirectional Lorenz CNN CDADS, nonautonomous bidirectional Chua CNN CCADS, and nonautonomously bidirectional Chen CNN CDADS and CCADS, respectively. Extensive numerical simulations show their complex dynamic behaviors. These theorems provide new means for understanding the GS phenomena of complex discrete and continuously differentiable networks.

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46

Sermone, Lelde. "Reduction of differentiable equations with impulse effect." Journal of Applied Mathematics and Stochastic Analysis 10, no. 1 (January 1, 1997): 79–87. http://dx.doi.org/10.1155/s1048953397000087.

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47

Papoulia, Katerina D. "Non-differentiable energy minimization for cohesive fracture." International Journal of Fracture 204, no. 2 (January 31, 2017): 143–58. http://dx.doi.org/10.1007/s10704-016-0167-x.

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48

Peng, Cheng, Haofu Liao, Gina Wong, Jiebo Luo, S. Kevin Zhou, and Rama Chellappa. "XraySyn: Realistic View Synthesis From a Single Radiograph Through CT Priors." Proceedings of the AAAI Conference on Artificial Intelligence 35, no. 1 (May 18, 2021): 436–44. http://dx.doi.org/10.1609/aaai.v35i1.16120.

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A radiograph visualizes the internal anatomy of a patient through the use of X-ray, which projects 3D information onto a 2D plane. Hence, radiograph analysis naturally requires physicians to relate their prior knowledge about 3D human anatomy to 2D radiographs. Synthesizing novel radiographic views in a small range can assist physicians in interpreting anatomy more reliably; however, radiograph view synthesis is heavily ill-posed, lacking in paired data, and lacking in differentiable operations to leverage learning-based approaches. To address these problems, we use Computed Tomography (CT) for radiograph simulation and design a differentiable projection algorithm, which enables us to achieve geometrically consistent transformations between the radiography and CT domains. Our method, XraySyn, can synthesize novel views on real radiographs through a combination of realistic simulation and finetuning on real radiographs. To the best of our knowledge, this is the first work on radiograph view synthesis. We show that by gaining an understanding of radiography in 3D space, our method can be applied to radiograph bone extraction and suppression without requiring groundtruth bone labels.

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Montes Maestre, Juan Sebastian, Yinwei Du, Ronan Hinchet, Stelian Coros, and Bernhard Thomaszewski. "Differentiable Stripe Patterns for Inverse Design of Structured Surfaces." ACM Transactions on Graphics 42, no. 4 (July 26, 2023): 1–14. http://dx.doi.org/10.1145/3592114.

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Stripe patterns are ubiquitous in nature and everyday life. While the synthesis of these patterns has been thoroughly studied in the literature, their potential to control the mechanics of structured materials remains largely unexplored. In this work, we introduce Differentiable Stripe Patterns---a computational approach for automated design of physical surfaces structured with stripe-shaped bi-material distributions. Our method builds on the work by Knöppel and colleagues [2015] for generating globally-continuous and equally-spaced stripe patterns. To unlock the full potential of this design space, we propose a gradient-based optimization tool to automatically compute stripe patterns that best approximate macromechanical performance goals. Specifically, we propose a computational model that combines solid shell finite elements with XFEM for accurate and fully-differentiable modeling of elastic bi-material surfaces. To resolve non-uniqueness problems in the original method, we furthermore propose a robust formulation that yields unique and differentiable stripe patterns. We combine these components with equilibrium state derivatives into an end-to-end differentiable pipeline that enables inverse design of mechanical stripe patterns. We demonstrate our method on a diverse set of examples that illustrate the potential of stripe patterns as a design space for structured materials. Our simulation results are experimentally validated on physical prototypes.

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50

Xing, Jiankai, Fujun Luan, Ling-Qi Yan, Xuejun Hu, Houde Qian, and Kun Xu. "Differentiable Rendering Using RGBXY Derivatives and Optimal Transport." ACM Transactions on Graphics 41, no. 6 (November 30, 2022): 1–13. http://dx.doi.org/10.1145/3550454.3555479.

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Traditional differentiable rendering approaches are usually hard to converge in inverse rendering optimizations, especially when initial and target object locations are not so close. Inspired by Lagrangian fluid simulation, we present a novel differentiable rendering method to address this problem. We associate each screen-space pixel with the visible 3D geometric point covered by the center of the pixel and compute derivatives on geometric points rather than on pixels. We refer to the associated geometric points as point proxies of pixels. For each point proxy, we compute its 5D RGBXY derivatives which measures how its 3D RGB color and 2D projected screen-space position change with respect to scene parameters. Furthermore, in order to capture global and long-range object motions, we utilize optimal transport based pixel matching to design a more sophisticated loss function. We have conducted experiments to evaluate the effectiveness of our proposed method on various inverse rendering applications and have demonstrated superior convergence behavior compared to state-of-the-art baselines.

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