Academic literature on the topic 'Neural fields equations'
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Journal articles on the topic "Neural fields equations"
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Veltz, Romain, and Olivier Faugeras. "A Center Manifold Result for Delayed Neural Fields Equations." SIAM Journal on Mathematical Analysis 45, no. 3 (2013): 1527–62. http://dx.doi.org/10.1137/110856162.
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Belhe, Yash, Michaël Gharbi, Matthew Fisher, Iliyan Georgiev, Ravi Ramamoorthi, and Tzu-Mao Li. "Discontinuity-Aware 2D Neural Fields." ACM Transactions on Graphics 42, no. 6 (2023): 1–11. http://dx.doi.org/10.1145/3618379.
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Neural image representations offer the possibility of high fidelity, compact storage, and resolution-independent accuracy, providing an attractive alternative to traditional pixel- and grid-based representations. However, coordinate neural networks fail to capture discontinuities present in the image and tend to blur across them; we aim to address this challenge. In many cases, such as rendered images, vector graphics, diffusion curves, or solutions to partial differential equations, the locations of the discontinuities are known. We take those locations as input, represented as linear, quadratic, or cubic Bézier curves, and construct a feature field that is discontinuous across these locations and smooth everywhere else. Finally, we use a shallow multi-layer perceptron to decode the features into the signal value. To construct the feature field, we develop a new data structure based on a curved triangular mesh, with features stored on the vertices and on a subset of the edges that are marked as discontinuous. We show that our method can be used to compress a 100, 000 2 -pixel rendered image into a 25MB file; can be used as a new diffusion-curve solver by combining with Monte-Carlo-based methods or directly supervised by the diffusion-curve energy; or can be used for compressing 2D physics simulation data.
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Nicks, Rachel, Abigail Cocks, Daniele Avitabile, Alan Johnston, and Stephen Coombes. "Understanding Sensory Induced Hallucinations: From Neural Fields to Amplitude Equations." SIAM Journal on Applied Dynamical Systems 20, no. 4 (2021): 1683–714. http://dx.doi.org/10.1137/20m1366885.
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Veltz, Romain, and Olivier Faugeras. "ERRATUM: A Center Manifold Result for Delayed Neural Fields Equations." SIAM Journal on Mathematical Analysis 47, no. 2 (2015): 1665–70. http://dx.doi.org/10.1137/140962279.
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Bressloff, Paul C., and Zachary P. Kilpatrick. "Nonlinear Langevin Equations for Wandering Patterns in Stochastic Neural Fields." SIAM Journal on Applied Dynamical Systems 14, no. 1 (2015): 305–34. http://dx.doi.org/10.1137/140990371.
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Scheinker, Alexander, and Reeju Pokharel. "Physics-constrained 3D convolutional neural networks for electrodynamics." APL Machine Learning 1, no. 2 (2023): 026109. http://dx.doi.org/10.1063/5.0132433.
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We present a physics-constrained neural network (PCNN) approach to solving Maxwell’s equations for the electromagnetic fields of intense relativistic charged particle beams. We create a 3D convolutional PCNN to map time-varying current and charge densities J(r, t) and ρ(r, t) to vector and scalar potentials A(r, t) and φ(r, t) from which we generate electromagnetic fields according to Maxwell’s equations: B = ∇ × A and E = −∇ φ − ∂A/ ∂t. Our PCNNs satisfy hard constraints, such as ∇ · B = 0, by construction. Soft constraints push A and φ toward satisfying the Lorenz gauge.
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Sim, Fabio M., Eka Budiarto, and Rusman Rusyadi. "Comparison and Analysis of Neural Solver Methods for Differential Equations in Physical Systems." ELKHA 13, no. 2 (2021): 134. http://dx.doi.org/10.26418/elkha.v13i2.49097.
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Differential equations are ubiquitous in many fields of study, yet not all equations, whether ordinary or partial, can be solved analytically. Traditional numerical methods such as time-stepping schemes have been devised to approximate these solutions. With the advent of modern deep learning, neural networks have become a viable alternative to traditional numerical methods. By reformulating the problem as an optimisation task, neural networks can be trained in a semi-supervised learning fashion to approximate nonlinear solutions. In this paper, neural solvers are implemented in TensorFlow for a variety of differential equations, namely: linear and nonlinear ordinary differential equations of the first and second order; Poisson’s equation, the heat equation, and the inviscid Burgers’ equation. Different methods, such as the naive and ansatz formulations, are contrasted, and their overall performance is analysed. Experimental data is also used to validate the neural solutions on test cases, specifically: the spring-mass system and Gauss’s law for electric fields. The errors of the neural solvers against exact solutions are investigated and found to surpass traditional schemes in certain cases. Although neural solvers will not replace the computational speed offered by traditional schemes in the near future, they remain a feasible, easy-to-implement substitute when all else fails.
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ITOH, MAKOTO, and LEON O. CHUA. "IMAGE PROCESSING AND SELF-ORGANIZING CNN." International Journal of Bifurcation and Chaos 15, no. 09 (2005): 2939–58. http://dx.doi.org/10.1142/s0218127405013794.
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CNN templates for image processing and pattern formation are derived from neural field equations, advection equations and reaction–diffusion equations by discretizing spatial integrals and derivatives. Many useful CNN templates are derived by this approach. Furthermore, self-organization is investigated from the viewpoint of divergence of vector fields.
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Wennekers, Thomas. "Dynamic Approximation of Spatiotemporal Receptive Fields in Nonlinear Neural Field Models." Neural Computation 14, no. 8 (2002): 1801–25. http://dx.doi.org/10.1162/089976602760128027.
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This article presents an approximation method to reduce the spatiotemporal behavior of localized activation peaks (also called “bumps”) in nonlinear neural field equations to a set of coupled ordinary differential equations (ODEs) for only the amplitudes and tuning widths of these peaks. This enables a simplified analysis of steady-state receptive fields and their stability, as well as spatiotemporal point spread functions and dynamic tuning properties. A lowest-order approximation for peak amplitudes alone shows that much of the well-studied behavior of small neural systems (e.g., the Wilson-Cowan oscillator) should carry over to localized solutions in neural fields. Full spatiotemporal response profiles can further be reconstructed from this low-dimensional approximation. The method is applied to two standard neural field models: a one-layer model with difference-of-gaussians connectivity kernel and a two-layer excitatory-inhibitory network. Similar models have been previously employed in numerical studies addressing orientation tuning of cortical simple cells. Explicit formulas for tuning properties, instabilities, and oscillation frequencies are given, and exemplary spatiotemporal response functions, reconstructed from the low-dimensional approximation, are compared with full network simulations.
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Mentzer, Katherine L., and J. Luc Peterson. "Neural network surrogate models for equations of state." Physics of Plasmas 30, no. 3 (2023): 032704. http://dx.doi.org/10.1063/5.0126708.
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Equation of state (EOS) data provide necessary information for accurate multiphysics modeling, which is necessary for fields such as inertial confinement fusion. Here, we suggest a neural network surrogate model of energy and entropy and use thermodynamic relationships to derive other necessary thermodynamic EOS quantities. We incorporate phase information into the model by training a phase classifier and using phase-specific regression models, which improves the modal prediction accuracy. Our model predicts energy values to 1% relative error and entropy to 3.5% relative error in a log-transformed space. Although sound speed predictions require further improvement, the derived pressure values are accurate within 10% relative error. Our results suggest that neural network models can effectively model EOS for inertial confinement fusion simulation applications.
Dissertations / Theses on the topic "Neural fields equations"
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Ueda, Hiroyuki. "Studies on low-field functional MRI to detect tiny neural magnetic fields." Doctoral thesis, Kyoto University, 2021. http://hdl.handle.net/2433/263666.
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付記する学位プログラム名: 京都大学卓越大学院プログラム「先端光・電子デバイス創成学」<br>京都大学<br>新制・課程博士<br>博士(工学)<br>甲第23205号<br>工博第4849号<br>京都大学大学院工学研究科電気工学専攻<br>(主査)教授 小林 哲生, 教授 松尾 哲司, 特定教授 中村 武恒<br>学位規則第4条第1項該当<br>Doctor of Philosophy (Engineering)<br>Kyoto University<br>DFAM
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Faye, Grégory. "Symmetry breaking and pattern formation in some neural field equations." Nice, 2012. http://www.theses.fr/2012NICE4017.
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Cette thèse se propose de comprendre la formation de structures dans les équations de champs neuronaux en présence de symétrie ainsi que la conséquence pour la modélisation du cortex visuel. Les équations de champs neuronaux sont des modèles mésoscopiques qui décrivent l'activité spatio-temporelle de populations de neurones. Elles ont été introduites dans les années 1970 et sont souvent appelées les équations de Wilson-Cowan-Amari en référence à leurs auteurs. D'un point de vue mathématique, les équations de champs neuronaux sont des équations intégro-différentielles posées sur des domaines qui dépendent des propriétés anatomiques et/ou fonctionnelles modélisées. Dans la première partie, nous rappelons quelques éléments de biologie du cortex visuel, dérivons les équations de champs neuronaux de manière générale et introduisons ensuite une nouvelle classe de champs neuronaux pour le problème de modélisation de la perception des textures. La seconde partie de cette thèse est dédiée à l'étude de formation de structures en géométrie non-euclidienne et s'appuie principalement sur la théorie des systèmes dynamiques en dimension infinie en présence de symétrie. Cette seconde partie est relativement indépendante des autres et est écrite de manière suffisamment générale pour pouvoir être appliquée de façon systématique à tout problème de formation de structures en géométrie non-euclidienne satisfaisant certaines conditions de généricité. Enfin, dans la dernière partie, nous étudions l'existence de solutions localisées pour une certaine classe de champs neuronaux définis sur des domaines non bornés<br>The aim of this Thesis is to give a deeper understanding of pattern formation in neural field equations with symmetry, and to understand the significance of these symmetries in modelling the visual cortex. Neural fields equations are mesoscopic models that describe the spatio-temporal activity of populations of neurons. They were introduced in the 1970s and are often called the Wilson-Cowan-Amari equations in reference to their authors. From a mathematical point of view, neural fields equations are integro-differential equations set on domains particular to the modelled anatomical / functional properties. The first part of the Thesis is an introduction to mesoscopic modelling of the visual cortex and presents a model of the processing of image edges and textures. The second part is dedicated to the study of spatially periodic solutions of neural field equations, in different geometries, with applications to visual hallucination patterns. The results developed are general enough to be applied to other pattern formation problems. Finally, the last part is centred on the study of localized solutions of neural field equations set on unbounded domains
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ALMEIDA, Arthur Santos de. "Algumas propriedades de equações diferenciais em espaços de Banach e aplicações de campos neurais." Universidade Federal de Campina Grande, 2015. http://dspace.sti.ufcg.edu.br:8080/jspui/handle/riufcg/1404.
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Previous issue date: 2015-08<br>Capes<br>Para ler o resumo deste trabalho recomendamos o download do arquivo, uma vez que o mesmo possui fórmulas e caracteres matemáticos que não foram possíveis trascreve-los aqui.<br>To read the summary of this work we recommend downloading the file, since it has formulas and mathematical characters that were not possible to transcribe them here.
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Wang, Wei 1974. "On solutions of advanced-retarded travelling wave equations arising in a neural field theory." Thesis, McGill University, 2006. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=98515.
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We consider a firing rate and a spike frequency adaptation (SFA) model of a one-dimensional neuronal network with axo-dendritic synaptic interactions. The neuronal network we focus on has a compactly supported connectivity pattern. The model equations are integro-differential equations with spatio-temporal delays arising from the finite axonal conduction velocity. We show that travelling wave solutions of these models are determined by mixed-type functional differential equations with both advanced and retarded delays. Especially, both delays depend on a unknown travelling wave speed. Based on the discussion of homogeneous steady states of model systems, we investigate the travelling front solutions of the firing rate model and the travelling pulse solutions of the SFA model. For the special case of a Heaviside firing rate function (the limit of infinite gain of a sigmoidal firing rate function), we obtain a special relationship between wave solutions, and then find analytic travelling wave solutions. For the models with a general sigmoidal firing rate function, we investigate the dynamic behavior in tails of travelling wave solutions, and perform the numerical calculation of travelling front solutions.<br>Keywords. Neuronal network; firing rate function; mixed-type functional differential equations; delays; travelling wave solutions
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Tamekue, Cyprien. "Controllability, Visual Illusions and Perception." Electronic Thesis or Diss., université Paris-Saclay, 2023. http://www.theses.fr/2023UPAST105.
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Cette thèse explore deux applications distinctes de la théorie du contrôle dans différents domaines scientifiques : la physique et les neurosciences. La première application se concentre sur la contrôlabilité nulle de l'équation parabolique associée à l'opérateur de Baouendi-Grushin sur la sphère de dimension 2. En revanche, la deuxième application concerne la description mathématique des illusions visuelles du type MacKay, et se focalise sur l'effet MacKay et les expériences psychophysiques de Billock et Tsou, via le contrôle de l'équation des champs neuronaux à une seule couche du type Amari. De plus, pour le besoin d'application à la stabilité entrée-état et la stabilisation robuste, la thèse examine l'existence d'un équilibre dans un modèle de population de champs neuronaux à plusieurs couches de Wilson-Cowan, plus précisément lorsque l'entrée sensorielle est un retour d'état proportionnelle agissant uniquement sur l'état des populations de neurones excitateurs.Dans la première partie, nous étudions les propriétés de contrôlabilité nulle de l'équation parabolique associée à l'opérateur de Baouendi-Grushin défini par la structure presque-riemannienne canonique sur la sphère bidimensionnelle. Cet opérateur présente une dégénérescence à l'équateur de la sphère. Nous fournissons certaines propriétés de contrôlabilité nulle de cette équation dans ce cadre courbé, ce qui généralise celles de l'équation parabolique de Baouendi-Grushin définie sur le plan.Concernant les neurosciences, dans un premier temps, on s'intéresse à la description des illusions visuelles pour lesquelles les outils de la théorie du contrôle et même de l'analyse multiéchelle semblent inappropriés.Dans notre discussion, nous utilisons l'équation des champs neuronaux de type Amari, dans laquelle l'entrée sensorielle est interprétée comme une représentation corticale du stimulus visuel utilisé dans chaque expérience. Elle contient une fonction de contrôle distribuée localisée qui modélise la spécificité du stimulus, par exemple, l'information redondante au centre du motif en entonnoir de MacKay (``rayons de MacKay'') ou le fait que les stimuli visuels dans les expériences de Billock et Tsou sont localisés dans le champ visuel.Toujours dans le cadre des neurosciences, nous étudions l'existence d'un équilibre dans un modèle de population de champs neuronaux à plusieurs couches de Wilson-Cowan lorsque l'entrée sensorielle est un retour d'état proportionnelle agissant uniquement sur l'état du système des populations de neurones excitateurs. Nous proposons une condition suffisante modérée sur les fonctions de réponse garantissant l'existence d'un tel point d'équilibre. L'intérêt de ce travail réside dans son application lors de l'étude de la pertubation des oscillations cérébrales pathologiques associées à la maladie de Parkinson lorsqu'on stimule et mesure uniquement la population de neurones excitateurs<br>This thesis explores two distinct control theory applications in different scientific domains: physics and neuroscience. The first application focuses on the null controllability of the parabolic, spherical Baouendi-Grushin equation. In contrast, the second application involves the mathematical description of the MacKay-type visual illusions, focusing on the MacKay effect and Billock and Tsou's psychophysical experiments by controlling the one-layer Amari-type neural fields equation. Additionally, intending to study input-to-state stability and robust stabilization, the thesis investigates the existence of equilibrium in a multi-layer neural fields population model of Wilson-Cowan, specifically when the sensory input is a proportional feedback acting only on the system's state of the populations of excitatory neurons.In the first part, we investigate the null controllability properties of the parabolic equation associated with the Baouendi-Grushin operator defined by the canonical almost-Riemannian structure on the 2-dimensional sphere. It presents a degeneracy at the equator of the sphere. We provide some null controllability properties of this equation to this curved setting, which generalize that of the parabolic Baouendi-Grushin equation defined on the plane.Regarding neuroscience, initially, the focus lies on the description of visual illusions for which the tools of bifurcation theory and even multiscale analysis appear unsuitable. In our study, we use the neural fields equation of Amari-type in which the sensory input is interpreted as a cortical representation of the visual stimulus used in each experiment. It contains a localised distributed control function that models the stimulus's specificity, e.g., the redundant information in the centre of MacKay's funnel pattern (``MacKay rays'') or the fact that visual stimuli in Billock and Tsou's experiments are localized in the visual field.Always within the framework of neurosciences, we investigate the existence of equilibrium in a multi-layers neural fields population model of Wilson-Cowan when the sensory input is a proportional feedback that acts only on the system's state of the population of excitatory neurons. There, we provide a mild condition on the response functions under which such an equilibrium exists. The interest of this work lies in its application in studying the disruption of pathological brain oscillations associated with Parkinson's disease when stimulating and measuring only the population of excitatory neurons
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SILVA, Michel Barros. "Comportamento Assintótico para Equação de Campos Neurais." Universidade Federal de Campina Grande, 2014. http://dspace.sti.ufcg.edu.br:8080/jspui/handle/riufcg/1395.
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MICHEL BARROS SILVA - DISSERTAÇÃO PPGMAT 2014..pdf: 335576 bytes, checksum: f2ee6b6d68cdefa6c32e300154d28756 (MD5)
Previous issue date: 2014-02<br>Capes<br>Para ler o reumo deste trabalho recomendamos o download do arquivo, pois o mesmo possui fórmulas e caracteres matemáticos que não foram possíveis transcreve-los.<br>To read the progress of this work we recommend downloading the file, as it has formulas and mathematical characters that could not be transcribed.
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Vellmer, Sebastian. "Applications of the Fokker-Planck Equation in Computational and Cognitive Neuroscience." Doctoral thesis, Humboldt-Universität zu Berlin, 2020. http://dx.doi.org/10.18452/21597.
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In dieser Arbeit werden mithilfe der Fokker-Planck-Gleichung die Statistiken, vor allem die Leistungsspektren, von Punktprozessen berechnet, die von mehrdimensionalen Integratorneuronen [Engl. integrate-and-fire (IF) neuron],
Netzwerken von IF Neuronen und Entscheidungsfindungsmodellen erzeugt werden.
Im Gehirn werden Informationen durch Pulszüge von Aktionspotentialen kodiert.
IF Neurone mit radikal vereinfachter Erzeugung von Aktionspotentialen haben sich in Studien die auf Pulszeiten fokussiert sind als Standardmodelle etabliert. Eindimensionale IF Modelle können jedoch beobachtetes Pulsverhalten oft nicht beschreiben und müssen dazu erweitert werden. Im erste Teil dieser Arbeit wird eine Theorie zur Berechnung der Pulszugleistungsspektren von stochastischen, multidimensionalen IF Neuronen entwickelt. Ausgehend von der zugehörigen Fokker-Planck-Gleichung werden partiellen Differentialgleichung abgeleitet, deren Lösung sowohl die stationäre Wahrscheinlichkeitsverteilung und Feuerrate, als auch das Pulszugleistungsspektrum beschreibt.
Im zweiten Teil wird eine Theorie für große, spärlich verbundene und homogene Netzwerke aus IF Neuronen entwickelt, in der berücksichtigt wird, dass die zeitlichen Korrelationen von Pulszügen selbstkonsistent sind. Neuronale Eingangströme werden durch farbiges Gaußsches Rauschen modelliert, das von einem mehrdimensionalen Ornstein-Uhlenbeck Prozess (OUP) erzeugt wird. Die Koeffizienten des OUP sind vorerst unbekannt und sind als Lösung der Theorie definiert. Um heterogene Netzwerke zu untersuchen, wird eine iterative Methode erweitert.
Im dritten Teil wird die Fokker-Planck-Gleichung auf Binärentscheidungen von Diffusionsentscheidungsmodellen [Engl. diffusion-decision models (DDM)] angewendet. Explizite Gleichungen für die Entscheidungszugstatistiken werden für den einfachsten und analytisch lösbaren Fall von der Fokker-Planck-Gleichung hergeleitet. Für nichtliniear Modelle wird die Schwellwertintegrationsmethode erweitert.<br>This thesis is concerned with the calculation of statistics, in particular the power spectra, of point processes generated by stochastic multidimensional integrate-and-fire (IF) neurons, networks of IF neurons and decision-making models from the corresponding Fokker-Planck equations.
In the brain, information is encoded by sequences of action potentials. In studies that focus on spike timing, IF neurons that drastically simplify the spike generation have become the standard model. One-dimensional IF neurons do not suffice to accurately model neural dynamics, however, the extension towards multiple dimensions yields realistic behavior at the price of growing complexity. The first part of this work develops a theory of spike-train power spectra for stochastic, multidimensional IF neurons. From the Fokker-Planck equation, a set of partial differential equations is derived that describes the stationary probability density, the firing rate and the spike-train power spectrum.
In the second part of this work, a mean-field theory of large and sparsely connected homogeneous networks of spiking neurons is developed that takes into account the self-consistent temporal correlations of spike trains. Neural input is approximated by colored Gaussian noise generated by a multidimensional Ornstein-Uhlenbeck process of which the coefficients are initially unknown but determined by the self-consistency condition and define the solution of the theory. To explore heterogeneous networks, an iterative scheme is extended to determine the distribution of spectra.
In the third part, the Fokker-Planck equation is applied to calculate the statistics of sequences of binary decisions from diffusion-decision models (DDM). For the analytically tractable DDM, the statistics are calculated from the corresponding Fokker-Planck equation. To determine the statistics for nonlinear models, the threshold-integration method is generalized.
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Daya, Bassam. "Résolution numérique des équations du champ neural : étude de la coordination du mouvement par des modèles mathématiques du cervelet." Angers, 1996. http://www.theses.fr/1996ANGE0013.
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Nous abordons le problème de la coordination du mouvement par les deux approches continues et discrètes, afin de les comparer en vue du neurocontrôle en robotique. Dans le premier chapitre, le formalisme des champs et les équations du champ pour un tissu nerveux ont été rappelés. Ces équations incluent les mécanismes physiologiques du système nerveux pour mieux tenir compte de la réalité. Les hypothèses permettant de retrouver les modèles classiques ont été déterminées, prouvant ainsi la généralité de la théorie du champ envisagée. Dans le deuxième chapitre, la résolution numérique des équations du champ est réalisée au moyen de la technique des éléments finis et des différences finies. Il est supposé que l'équation au niveau neuronal est bidimensionnelle, alors que celle du niveau synaptique est à une dimension. A la fin de ce chapitre, nous proposons des simulations pour les réseaux multicouches classiques illustrant la propagation de l'activation. Dans le troisième chapitre, nous présentons les premiers résultats de l'application du formalisme des champs au cervelet. En particulier, les propriétés concernant l'effet de la localisation géométrique des neurones et l'effet de la hiérarchie sont déduites. Dans le quatrième chapitre, nous commençons par l'étude du modèle du cervelet en boucle fermée en analysant le rôle de la fibre grimpante dont on sait qu'elle véhicule le signal d'erreur. La méthode analytique utilisée, est fondée sur une technique classique d'optimisation. Nous généralisons ensuite notre méthode. Enfin nous abordons dans une dernière section un aspect de la robotique mobile. Nous avons considéré l'exemple simplifie d'un bipède dont il faut contrôler l'équilibre dynamique par l'accélération articulaire du tronc. On montre que notre modèle permet d'apprendre à contrôler l'équilibre dynamique du système pour les trajectoires apprises et d'anticiper celles qui sont non apprises.
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Quininao, Cristobal. "Mathematical modeling in neuroscience : collective behavior of neuronal networks & the role of local homeoproteins diffusion in morphogenesis." Thesis, Paris 6, 2015. http://www.theses.fr/2015PA066152/document.
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Ce travail est consacré à l’étude de quelques questions issues de la modélisation des systèmes biologiques en combinant des outils analytiques et probabilistes. Dans la première partie, nous nous intéressons à la dérivation des équations de champ moyen associées aux réseaux de neurones, ainsi qu’à l’étude de la convergence vers l’équilibre des solutions. Dans le Chapitre 2, nous utilisons la méthode de couplage pour démontrer la propagation du chaos pour un réseau neuronal avec délais et avec une architecture aléatoire. Dans le Chapitre 3, nous considérons une équation cinétique du type FitzHugh-Nagumo. Nous analysons l'existence de solutions et prouvons la convergence exponentielle dans les régimes de faible connectivité. Dans la deuxième partie, nous étudions le rôle des homéoprotéines (HPs) sur la robustesse des bords des aires fonctionnelles. Dans le Chapitre 4, nous proposons un modèle général du développement neuronal. Nous prouvons qu'en l'absence de diffusion, les HPs sont exprimées dans des régions irrégulières. Mais en présence de diffusion, même arbitrairement faible, des frontières bien définies émergent. Dans le Chapitre 5, nous considérons le modèle général dans le cas unidimensionnel et prouvons l'existence de solutions stationnaires monotones définissant un point d'intersection unique aussi faible que soit le coefficient de diffusion. Enfin, dans la troisième partie, nous étudions une équation de Keller-Segel sous-critique. Nous démontrons la propagation du chaos sans aucune restriction sur le noyau de force. En outre, nous démontrons que la propagation du chaos a lieu dans le sens de l’entropie<br>This work is devoted to the study of mathematical questions arising from the modeling of biological systems combining analytic and probabilistic tools. In the first part, we are interested in the derivation of the mean-field equations related to some neuronal networks, and in the study of the convergence to the equilibria of the solutions to the limit equations. In Chapter 2, we use the coupling method to prove the chaos propagation for a neuronal network with delays and random architecture. In Chapter 3, we consider a kinetic FitzHugh-Nagumo equation. We analyze the existence of solutions and prove the nonlinear exponential convergence in the weak connectivity regime. In the second part, we study the role of homeoproteins (HPs) on the robustness of boundaries of functional areas. In Chapter 4, we propose a general model for neuronal development. We prove that in the absence of diffusion, the HPs are expressed on irregular areas. But in presence of diffusion, even arbitrarily small, well defined boundaries emerge. In Chapter 5, we consider the general model in the one dimensional case and prove the existence of monotonic stationary solutions defining a unique intersection point for any arbitrarily small diffusion coefficient. Finally, in the third part, we study a subcritical Keller-Segel equation. We show the chaos propagation without any restriction on the force kernel. Eventually, we demonstrate that the propagation of chaos holds in the entropic sense
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Костів, Б. В. "Удосконалення безкоштовного визначення струмів в стінках підземних трубопроводів для контролю їх ізоляційного покриття". Thesis, Івано-Франківський національний технічний університет нафти і газу, 2010. http://elar.nung.edu.ua/handle/123456789/1974.
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У дисертації розроблено спосіб безконтактного визначення струму в стінках одного підземного трубопроводу на основі однократного вимірювання напруженостей п’ятьма магнітними антенами, що знаходяться в двох блоках, без попередньої орієнтації бази вимірювальної системи в перпендикулярній до осі трубопроводу площині. Розроблено спосіб автоматичного профілювання горизонтальної складової напруженості магнітного поля при проходженні із вимірювачьною системою над трубопроводами в перпендикулярному відносно їх осей напрямку. Запропоновано використання трьохшарової нейронної мережі для безконтактного визначення струму в стінках одного і двох підземних трубопроводів на основі даних профілю напруженостей магнітного поля над цими трубопроводами. Розроблено спосіб, в якому передбачено використання умовних рівнянь і отримання на їх базі нормальних рівнянь для безконтактного визначення струмів в стінках підземних трубопроводів при перпендикулярному проходженні над ними. Запропоновано структурну схему і виготовлено систему для безконтактного визначення струму в стінках підземних трубопроводів, яка реалізує всі запропоновані способи визначення цих струмів. Виконано метрологічний аналіз
розробленої системи безконтактного визначення струмів в підземних трубопроводах, розроблена установка, яка дає змогу проводити експериментальні дослідження метрологічних характеристик розробленої системи безконтактного визначення струмів в підземних трубопроводах, а також подібних їй приладів і систем. Визначено
метрологічні показники розробленої системи при безконтактному визначенні струмів у стінках контрольованих трубопроводів. Проведені лабораторні, польові і промислові випробування розробленої системи, які підтвердили її працездатність і можливість використання для контролю ізоляційного покриття підземних трубопроводів на основі заникання струму вздовж траси.<br>У дисертації розроблено спосіб безконтактного визначення струму в стінках одного підземного трубопроводу на основі однократного вимірювання напруженостей п’ятьма магнітними антенами, що знаходяться в двох блоках, без попередньої орієнтації бази вимірювальної системи в перпендикулярній до осі трубопроводу площині. Розроблено спосіб автоматичного профілювання горизонтальної складової напруженості магнітного поля при проходженні із вимірювачьною системою над трубопроводами в перпендикулярному відносно їх осей напрямку. Запропоновано використання трьохшарової нейронної мережі для безконтактного визначення струму в стінках одного і двох підземних трубопроводів на основі даних профілю напруженостей магнітного поля над цими трубопроводами. Розроблено спосіб, в якому передбачено використання умовних рівнянь і отримання на їх базі нормальних рівнянь для безконтактного визначення струмів в стінках підземних трубопроводів при перпендикулярному проходженні над ними. Запропоновано структурну схему і виготовлено систему для безконтактного визначення струму в стінках підземних трубопроводів, яка реалізує всі запропоновані способи визначення цих струмів. Виконано метрологічний аналіз
розробленої системи безконтактного визначення струмів в підземних трубопроводах, розроблена установка, яка дає змогу проводити експериментальні дослідження метрологічних характеристик розробленої системи безконтактного визначення струмів в підземних трубопроводах, а також подібних їй приладів і систем. Визначено
метрологічні показники розробленої системи при безконтактному визначенні струмів у стінках контрольованих трубопроводів. Проведені лабораторні, польові і промислові випробування розробленої системи, які підтвердили її працездатність і можливість використання для контролю ізоляційного покриття підземних трубопроводів на основі заникання струму вздовж траси.
Books on the topic "Neural fields equations"
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Potthast, Roland, P. Beim Graben, Wright James, and Stephen Coombes. Neural Fields: Theory and Applications. Springer Berlin / Heidelberg, 2016.
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Potthast, Roland, Wright James, Stephen Coombes, and Peter beim Graben. Neural Fields: Theory and Applications. Springer London, Limited, 2014.
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Waves In Neural Media From Single Neurons To Neural Fields. Springer-Verlag New York Inc., 2013.
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Wadman, Wytse J., and Fernando H. Lopes da Silva. Biophysical Aspects of EEG and MEG Generation. Edited by Donald L. Schomer and Fernando H. Lopes da Silva. Oxford University Press, 2017. http://dx.doi.org/10.1093/med/9780190228484.003.0004.
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This chapter reviews the essential physical principles involved in the generation of electroencephalographic (EEG) and magnetoencephalographic (MEG) signals. The general laws governing the electrophysiology of neuronal activity are analyzed within the formalism of the Maxwell equations that constitute the basis for understanding electromagnetic fields in general. Three main topics are discussed. The first is the forward problem: How can one calculate the electrical field that results from a known configuration of neuronal sources? The second is the inverse problem: Given an electrical field as a function of space and time mostly recorded at the scalp (EEG/MEG), how can one reconstruct the underlying generators at the brain level? The third is the reverse problem: How can brain activity be modulated by external electromagnetic fields with diagnostic and/or therapeutic objectives? The chapter emphasizes the importance of understanding the common biophysical framework concerning these three main topics of brain electrical and magnetic activities.
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van Hinsbergh, Victor W. M. Physiology of blood vessels. Oxford University Press, 2017. http://dx.doi.org/10.1093/med/9780198755777.003.0002.
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This chapter covers two major fields of the blood circulation: ‘distribution’ and ‘exchange’. After a short survey of the types of vessels, which form the circulation system together with the heart, the chapter describes how hydrostatic pressure derived from the heartbeat and vascular resistance determine the volume of blood that is locally delivered per time unit. The vascular resistance depends on the length of the vessel, blood viscosity, and, in particular, on the diameter of the vessel, as formulated in the Poiseuille-Hagen equation. Blood flow can be determined in vivo by different imaging modalities. A summary is provided of how smooth muscle cell contraction is regulated at the cellular level, and how neuronal, humoral, and paracrine factors affect smooth muscle contraction and thereby blood pressure and blood volume distribution among tissues. Subsequently the exchange of solutes and macromolecules over the capillary endothelium and the contribution of its surface layer, the glycocalyx, are discussed. After a description of the Starling equation for capillary exchange, new insights are summarized(in the so-called glycocalyx cleft model) that led to a new view on exchange along the capillary and on the contribution of oncotic pressure. Finally mechanisms are indicated in brief that play a role in keeping the blood volume constant, as a constant volume is a prerequisite for adequate functioning of the circulatory system.
Book chapters on the topic "Neural fields equations"
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Helias, Moritz, and David Dahmen. "Functional Formulation of Stochastic Differential Equations." In Statistical Field Theory for Neural Networks. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-46444-8_7.
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Helias, Moritz, and David Dahmen. "Perturbation Theory for Stochastic Differential Equations." In Statistical Field Theory for Neural Networks. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-46444-8_9.
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Alecu, Lucian, and Hervé Frezza-Buet. "Application-Driven Parameter Tuning Methodology for Dynamic Neural Field Equations." In Neural Information Processing. Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-10677-4_15.
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La Camera, Giancarlo. "The Mean Field Approach for Populations of Spiking Neurons." In Advances in Experimental Medicine and Biology. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-89439-9_6.
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AbstractMean field theory is a device to analyze the collective behavior of a dynamical system comprising many interacting particles. The theory allows to reduce the behavior of the system to the properties of a handful of parameters. In neural circuits, these parameters are typically the firing rates of distinct, homogeneous subgroups of neurons. Knowledge of the firing rates under conditions of interest can reveal essential information on both the dynamics of neural circuits and the way they can subserve brain function. The goal of this chapter is to provide an elementary introduction to the mean field approach for populations of spiking neurons. We introduce the general idea in networks of binary neurons, starting from the most basic results and then generalizing to more relevant situations. This allows to derive the mean field equations in a simplified setting. We then derive the mean field equations for populations of integrate-and-fire neurons. An effort is made to derive the main equations of the theory using only elementary methods from calculus and probability theory. The chapter ends with a discussion of the assumptions of the theory and some of the consequences of violating those assumptions. This discussion includes an introduction to balanced and metastable networks and a brief catalogue of successful applications of the mean field approach to the study of neural circuits.
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La Camera, Giancarlo. "The Mean Field Approach for Populations of Spiking Neurons." In Advances in Experimental Medicine and Biology. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-89439-9_6.
Full textAbstract:
AbstractMean field theory is a device to analyze the collective behavior of a dynamical system comprising many interacting particles. The theory allows to reduce the behavior of the system to the properties of a handful of parameters. In neural circuits, these parameters are typically the firing rates of distinct, homogeneous subgroups of neurons. Knowledge of the firing rates under conditions of interest can reveal essential information on both the dynamics of neural circuits and the way they can subserve brain function. The goal of this chapter is to provide an elementary introduction to the mean field approach for populations of spiking neurons. We introduce the general idea in networks of binary neurons, starting from the most basic results and then generalizing to more relevant situations. This allows to derive the mean field equations in a simplified setting. We then derive the mean field equations for populations of integrate-and-fire neurons. An effort is made to derive the main equations of the theory using only elementary methods from calculus and probability theory. The chapter ends with a discussion of the assumptions of the theory and some of the consequences of violating those assumptions. This discussion includes an introduction to balanced and metastable networks and a brief catalogue of successful applications of the mean field approach to the study of neural circuits.
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Lima, Pedro M. "Numerical Investigation of Stochastic Neural Field Equations." In Advances in Mechanics and Mathematics. Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-02487-1_2.
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Takabe, Hideaki. "Basic Properties of Plasma in Fluid Model." In Springer Series in Plasma Science and Technology. Springer International Publishing, 2024. http://dx.doi.org/10.1007/978-3-031-45473-8_2.
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AbstractIf the spatial variation of plasma is longer than the particle mean free path and the time variation is sufficiently longer than the plasma Coulomb collision time, the plasma can be approximated as being in local thermal equilibrium (LTE) at any point (t, r). Then the velocity distribution functions of the particles become Maxwellian. In addition, assuming Maxwellian is also a good assumption in many cases even for collisionless plasmas such as high-temperature fusion plasmas. In the fluid model of plasmas, The plasmas can be described in terms of five variables characterizing local Maxwellian: the density n(t, r), flow velocity vector u(t,r), and temperature T(t, r). So, the mathematics used in fluid physics is widely applicable to studying plasma phenomena.Although conventional fluids are neutral, plasma fluids of electrons and ions couple with electromagnetic fields. It is, therefore, necessary to solve Maxwell’s equations simultaneously. It is also possible to approximate electrons and ions as two different fluids or as a single fluid in case-by-case. This requires an insight into what kind of physics is important in our problem.After reviewing the basic equation of fluids, several fluid models for plasmas are shown. Especially, a variety of waves appears because of charged particle fluids are derived to know why waves are fundamental to knowing the plasma dynamics. The mathematical method to obtain the wave solutions as an initial value problem is explained as well as the meaning of the resultant dispersion relations.Magneto-hydrodynamic equations (MHD) are derived to explain the effects of the Biermann battery, magnetic dynamo, etc. The relationship of magnetic field and vortex flow is studied. Resistive MHD is derived including the Nernst effect, which becomes important for the magnetic field in strong electron heat flux.Finally, electromagnetic (EM) waves in magnetized plasmas are derived to see how to use for diagnostics in the laboratory and observation of wide range of electromagnetic waves from the Universe.
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Burlakov, Evgenii, Vitaly Verkhlyutov, Ivan Malkov, and Vadim Ushakov. "Assessment of Cortical Travelling Waves Parameters Using Radially Symmetric Solutions to Neural Field Equations with Microstructure." In Advances in Neural Computation, Machine Learning, and Cognitive Research IV. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-60577-3_5.
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Takabe, Hideaki. "Non-local Transport of Electrons in Plasmas." In Springer Series in Plasma Science and Technology. Springer International Publishing, 2024. http://dx.doi.org/10.1007/978-3-031-45473-8_6.
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AbstractSince plasma is high temperature and the charge particles are running with high temperature, for example, at 1 keV, about the velocity of 109 (electron) and 2 × 107 (ion) [cm/s]. Since Coulomb mean-free-path is proportional to (velocity)4, higher velocity component transfers its energy over a long distance without Coulomb collision. This is usually called as “non-local transport” and the traditional diffusion model in neutral gas cannot be applicable. In laser plasma, the locally heated electron thermal energy is transported into cold over-dense region non-locally. The best way to solve such problem is to solve Fokker-Planck equation, while it is time consuming and some theoretical models have been proposed and studied over the last four decades. The physics of such models are explained here and most recent model SNB is shown and compared to experiments. The difficulty of transport of charges particles such as electrons is how to include the effect of electrostatic field and magnetic field self-consistently.
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Haskell, Evan C., and Vehbi E. Paksoy. "Localized Activity States for Neuronal Field Equations of Feature Selectivity in a Stimulus Space with Toroidal Topology." In Nonlinear and Complex Dynamics. Springer New York, 2011. http://dx.doi.org/10.1007/978-1-4614-0231-2_17.
Full textConference papers on the topic "Neural fields equations"
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Yan, Chang, Shengjun Ju, Dilong Guo, Guowei Yang, and Shuanbao Yao. "Inferring Unsteady Wake Flow Fields From Partial Data by Physics-Informed Neural Networks." In ASME 2022 Fluids Engineering Division Summer Meeting. American Society of Mechanical Engineers, 2022. http://dx.doi.org/10.1115/fedsm2022-86945.
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Abstract Massive differential numerical computations are necessary in Computational Fluid Dynamics. In addition, the experimental results are generally noisy. Consequently, traditional methods cannot get unsteady flow fields immediately and precisely. In this research, the inferences of unsteady wake flow fields at different Reynolds numbers by Physics-Informed Neural Networks (PINNs) are studied. Unlike typical neural networks whose loss function consists of Mean Square Error only, the loss function of PINNs consists of Mean Square Error and the sum of squares of residuals of the flow governing equations. The flow governing equations are introduced to the neural networks as a regularization of the loss function. The existence of regular term reduces the dependence on labeled data during training. Then the PINNs is trained with very little labeled data (5% of the full field). After being trained, the PINNs show excellent performance in inferring the unsteady wake flow fields. When the Reynolds number is 1e2, the Mean Absolute Error (MAE) of the reconstructed velocity field is on the order of 1e−4. Meanwhile, the MAE increases with the increase of Reynolds number. In addition, even if the random noise of the training set is introduced up to 20%, the result is still acceptable, which demonstrates the great anti-noise ability of physics-informed neural networks.
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Jo, Minju, Seungji Kook, and Noseong Park. "Hawkes Process Based on Controlled Differential Equations." In Thirty-Second International Joint Conference on Artificial Intelligence {IJCAI-23}. International Joint Conferences on Artificial Intelligence Organization, 2023. http://dx.doi.org/10.24963/ijcai.2023/239.
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Hawkes processes are a popular framework to model the occurrence of sequential events, i.e., occurrence dynamics, in several fields such as social diffusion. In real-world scenarios, the inter-arrival time among events is irregular. However, existing neural network-based Hawkes process models not only i) fail to capture such complicated irregular dynamics, but also ii) resort to heuristics to calculate the log-likelihood of events since they are mostly based on neural networks designed for regular discrete inputs. To this end, we present the concept of Hawkes process based on controlled differential equations (HP-CDE), by adopting the neural controlled differential equation (neural CDE) technology which is an analogue to continuous RNNs. Since HP-CDE continuously reads data, i) irregular time-series datasets can be properly treated preserving their uneven temporal spaces, and ii) the log-likelihood can be exactly computed. Moreover, as both Hawkes processes and neural CDEs are first developed to model complicated human behavioral dynamics, neural CDE-based Hawkes processes are successful in modeling such occurrence dynamics. In our experiments with 4 real-world datasets, our method outperforms existing methods by non-trivial margins.
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Zhang, Chi, Shihao Wang, and Yu-Shu Wu. "A Physics-Informed Neural Network for Temporospatial Prediction of Hydraulic-Geomechanical Processes." In SPE Reservoir Simulation Conference. SPE, 2023. http://dx.doi.org/10.2118/212202-ms.
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Abstract This work aims to quantify the temporal and spatial evolution of pressure and stress fields in poroelastic reservoirs by replacing the conventional reservoir-geomechanical simulators with a novel convolutional-recurrent network (CNN-RNN) proxy. The proposed convolutional-recurrent neural network uses the governing equations of the coupled hydraulic-geomechanical process as the loss function. Initial conditions and spatial rock property fields are taken as inputs to predict the variation of pressure and stress fields. A customized convolutional filter mimicking the higher-order finite difference approach is adopted to improve the solution accuracy of the network. We apply the neural network to solve one synthetic 2D hydraulic-geomechanical problem. The pressure and stress fields predicted from our neural network are compared with the reference numerical solutions derived from the finite difference method. The performance exhibits the potential of the proposed deep learning model for hydraulic-geomechanical processes simulation. The predicted pressure field displays a high degree of accuracy up to 95%, while the error in stress prediction is slightly higher due to the limitation of the current adopted neural network. In particular, our model outperforms the traditional second-order finite difference method in both speed and accuracy. Overall, the work shows the capability of the neural network to capture temporospatial prediction in hydraulic-geomechanical processes.
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Post, Pascal, Benjamin Winhart, and Francesca di Mare. "Investigation of Physics-Informed Neural Networks Based Solution Techniques for Internal Flows." In ASME Turbo Expo 2022: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2022. http://dx.doi.org/10.1115/gt2022-80960.
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Abstract In this work, we explore for the first time the possibility and potentials of employing the emerging PINNs approach in internal flow configurations, solving the steady state Euler equations in two dimensions for forward and inverse problems. In addition to a simple bump test case, the PINNs results of a highly loaded transonic linear turbine guide vane cascade are presented. For forward problems, we investigate different formulations of the transport equations and boundary conditions. Overall, PINNs approximate the solution with acceptable accuracy; however, conventional CFD methods are far superior in forward settings. Finally, we demonstrate the capabilities and the tremendous potentials of PINNs regarding hidden fluid mechanics in two distinct inverse settings, intractable for conventional CFD methods. Firstly, we infer complete flow fields based on partial, possible noisy, solution data, e.g., partial surface pressure and velocity field data; even approximating the exit condition of the cascade using only the measured blade pressure distribution is possible. Secondly, we also infer an unknown parameter of the governing equations.
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Wang, Jun, Kevin Chiu, and Mark Fuge. "Learning to Abstract and Compose Mechanical Device Function and Behavior." In ASME 2020 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/detc2020-22714.
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Abstract While current neural networks (NNs) are becoming good at deriving single types of abstractions for a small set of phenomena, for example, using a single NN to predict a flow velocity field, NNs are not good at composing large systems as compositions of small phenomena and reasoning about their interactions. We want to study how NNs build both the abstraction and composition of phenomena when a single NN model cannot suffice. Rather than a single NN that learns one physical or social phenomenon, we want a group of NNs that learn to abstract, compose, reason, and correct the behaviors of different parts in a system. In this paper, we investigate the joint use of Physics-Informed (Navier-Stokes equations) Deep Neural Networks (i.e., Deconvolutional Neural Networks) as well as Geometric Deep Learning (i.e., Graph Neural Networks) to learn and compose fluid component behavior. Our models successfully predict the fluid flows and their composition behaviors (i.e., velocity fields) with an accuracy of about 99%.
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Ye, Ximeng, Hongyu Li, and Guoliang Qin. "Solving Flows Across Rotor and Stator Cascades With Local Neural Operator for Computational Fluid Dynamics." In ASME 2023 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2023. http://dx.doi.org/10.1115/imece2023-116339.
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Abstract This work presents a practical application of local neural operator (LNO) in computational fluid dynamics (CFD), which is a promising newly proposed deep learning based numerical solver for transient partial differential equations. Here, LNO is trained to learn the intrinsic law behind compressible Navier-Stokes equations and then solve arbitrary unseen flows by collaborating with specifically designed boundary treatment. With LNO as a CFD method, we investigate the unsteady flow across cascades of airfoils in turbomachinery. According to the results, LNO correctly and efficiently captures the fluctuations of velocity fields due to the interaction between the rotor and stator. The mean square error for velocity is below 0.082 and the computation is ×169 faster than conventional finite element method. We further conduct a brief investigation on several cases with different arrangements of cascades. These practices show great flexibility of LNO in tasks with complex geometries, diverse boundaries, and delicate flow patterns, indicating great potential for applications in various scientific research and engineering.
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Nagy, Allen L. "Individual differences in color discrimination and neural coding." In OSA Annual Meeting. Optica Publishing Group, 1986. http://dx.doi.org/10.1364/oam.1986.my4.
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Individual differences in color discrimination are analyzed in terms of three dimensions that differ from both receptoral and perceptual dimensions. These three dimensions may be described as achromatic, red/green, and tritan axes. When experimental measures of color difference thresholds, or discrimination contours derived from color matching data, are analyzed in terms of these three axes, much of the differences between individuals with normal color vision can be accounted for by individual differences in sensitivity on each of the axes. Thresholds along each axis vary regularly as a function of chromaticity and luminance and may be described by equations, but different parameters are required for different individuals. In fixed neutral conditions, sensitivities along the two chromatic axes may be significantly correlated across normal observers, but these may not be highly correlated with sensitivity along the achromatic axis. With prolonged viewing, the color difference threshold for two adjacent fields may increase.
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Alhubail, Ali, Marwan Fahs, Francois Lehmann, and Hussein Hoteit. "Physics-Informed Neural Networks for Modeling Flow in Heterogeneous Porous Media: A Decoupled Pressure-Velocity Approach." In International Petroleum Technology Conference. IPTC, 2024. http://dx.doi.org/10.2523/iptc-24362-ms.
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Abstract Physics-informed neural networks (PINNs) have shown success in solving physical problems in various fields. However, PINNs face major limitations when addressing fluid flow in heterogeneous porous media, related to discontinuities in rock properties. This is because automatic differentiation is inadequate for evaluating the spatial derivatives of hydraulic conductivity where it is discontinuous. This study aims to devise PINN implementations that overcome this limitation. This work proposes decoupling the mass conservation equation from Darcy's law and utilizing the residuals of these decoupled equations to train the loss function of the PINN, rather than using a single residual from the combined equation. As a result, we circumvent the need to find the spatial derivative of the discontinuous hydraulic conductivity, and instead, we impose the continuity of fluxes. This decoupling necessitates that each primary unknown (pressure and velocity components) be computed by the neural networks (NNs) rather than deriving the velocity (or fluxes) from the pressure. We examined three NN configurations and compared their performance by analyzing their accuracy and training time for various 2D scenarios. These scenarios explored various boundary conditions, different hydraulic conductivity fields, as well as different orientations of the heterogeneous media within the domain of interest. In these problems, the pressure and velocity field are the primary unknowns. The three configurations include: (a) one NN with the three unknowns as its outputs, (b) two NNs, one outputting pressure and the other outputting the velocity, and (c) three NNs, each having one primary unknown as an output. Utilizing these NN architectures, we were able to solve the heterogeneous problems with varying levels of accuracy when compared to results from numerical simulators. While maintaining a similar number of training parameters for a fair assessment, the configuration with three NNs yielded the most accurate results, with a comparable training time to the other configurations. Using this optimal configuration, we performed a sensitivity analysis to demonstrate the effect of modifying the NN(s) hyperparameters, such as the number of layers, the number of nodes per layer, and the learning rate, on the accuracy of the results. We introduce a novel PINN approach for modeling fluid flow in heterogeneous media. This proposed method not only preserves the inherent discontinuity of rock petrophysical properties but also leverages the benefits of automatic differentiation. By incorporating this PINN architecture, we have opened up new possibilities for extending the application of PINN to realistic reservoir simulations that capture the complexities of the subsurface.
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Cai, Shengze, Zhicheng Wang, Chryssostomos Chryssostomidis, and George Em Karniadakis. "Heat Transfer Prediction With Unknown Thermal Boundary Conditions Using Physics-Informed Neural Networks." In ASME 2020 Fluids Engineering Division Summer Meeting collocated with the ASME 2020 Heat Transfer Summer Conference and the ASME 2020 18th International Conference on Nanochannels, Microchannels, and Minichannels. American Society of Mechanical Engineers, 2020. http://dx.doi.org/10.1115/fedsm2020-20159.
Full textAbstract:
Abstract Simulating convective heat transfer using traditional numerical methods requires explicit definition of thermal boundary conditions on all boundaries of the domain, which is almost impossible to fulfill in real applications. Here, we address this ill-posed problem using machine learning techniques by assuming that we have some extra measurements of the temperature at a few locations in the domain, not necessarily located on the boundaries with the unknown thermal boundary condition. In particular, we employ physics-informed neural networks (PINNs) to represent the velocity and temperature fields while simultaneously enforce the Navier-Stokes and energy equations at random points in the domain. In PINNs, all differential operators are computed using automatic differentiation, hence avoiding discretization in either space or time. The loss function is composed of multiple terms, including the mismatch in the velocity and temperature data, the boundary and initial conditions, as well as the residuals of the Navier-Stokes and energy equations. Here, we develop a data-driven strategy based on PINNs to infer the temperature field in the prototypical problem of convective heat transfer in flow past a cylinder. We assume that we have just a couple of temperature measurements on the cylinder surface and a couple more temperature measurements in the wake region, but the thermal boundary condition on the cylinder surface is totally unknown. Upon training the PINN, we can discover the unknown boundary condition while simultaneously infer the temperature field everywhere in the domain with less than 5% error in the Nusselt number prediction. In order to assess the performance of PINN, we carried out a high fidelity simulation of the same heat transfer problem (with known thermal boundary conditions) by using the high-order spectral/hp-element method (SEM), and quantitatively evaluated the accuracy of PINN’s prediction with respect to SEM. We also propose a method to adaptively select the location of sensors in order to minimize the number of required temperature measurements while increasing the accuracy of the inference in heat transfer.
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Taraghi, Pouya, Yong Li, Nader Yoosef-Ghodsi, Matt Fowler, Muntaseer Kainat, and Samer Adeeb. "Response of Buried Pipelines Under Permanent Ground Movements: Physics-Informed Deep Neural Network Approach." In ASME 2023 Pressure Vessels & Piping Conference. American Society of Mechanical Engineers, 2023. http://dx.doi.org/10.1115/pvp2023-106201.
Full textAbstract:
Abstract Long-distance underground pipelines are inexorably constructed over harsh geological fields exposing them to geohazards resulting in both transient and permanent ground movement. The permanent ground movements caused by landslides, liquefaction, ground subsidence, slope failures, fault movement, etc., can result in large deformation threatening the safety and integrity of the pipelines. Therefore, evaluation of the structural response, i.e., displacement and strain fields of the buried pipelines exposed to the permanent ground displacement is a major concern for the industry. The mechanical and physical behavior of the pipelines subjected to ground movement can be described mathematically using nonlinear Partial Differential Equations (PDEs) by adopting the Euler-Bernoulli beam theory with large deformation. The model takes into account the nonlinearities caused by the pipe-soil interaction and pipe geometry. Various methods, including the Finite Element Method (FEM) and Finite Difference Method (FDM), can be used to solve the nonlinear PDEs and thus predict the structural response. However, the FEM usually requires complicated simulation using costly commercial software. Also, both the FE and FD methods are mesh-dependent. On this basis, this research endeavor proposes and utilizes an inexpensive, novel, easy-to-implemented, simulation-free, and meshless method to get the response. As a result, Physics-Informed Neural Networks (PINNs), deep-learning neural networks that consider the underlying law of physics in PDEs, can be a potential strategy to deal with PDEs with high complexity. To this end, the neural networks thoroughly learn based on the incorporated physical laws together with the boundary and/or initial conditions, eliminating the need for large training datasets. The obtained structural response, applicability, and accuracy of the predicted results of the proposed method are assessed by comparison with the obtained results from FEM.
Reports on the topic "Neural fields equations"
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Warrick, Arthur W., Gideon Oron, Mary M. Poulton, Rony Wallach, and Alex Furman. Multi-Dimensional Infiltration and Distribution of Water of Different Qualities and Solutes Related Through Artificial Neural Networks. United States Department of Agriculture, 2009. http://dx.doi.org/10.32747/2009.7695865.bard.
Full textAbstract:
The project exploits the use of Artificial Neural Networks (ANN) to describe infiltration, water, and solute distribution in the soil during irrigation. It provides a method of simulating water and solute movement in the subsurface which, in principle, is different and has some advantages over the more common approach of numerical modeling of flow and transport equations. The five objectives were (i) Numerically develop a database for the prediction of water and solute distribution for irrigation; (ii) Develop predictive models using ANN; (iii) Develop an experimental (laboratory) database of water distribution with time; within a transparent flow cell by high resolution CCD video camera; (iv) Conduct field studies to provide basic data for developing and testing the ANN; and (v) Investigate the inclusion of water quality [salinity and organic matter (OM)] in an ANN model used for predicting infiltration and subsurface water distribution. A major accomplishment was the successful use of Moment Analysis (MA) to characterize “plumes of water” applied by various types of irrigation (including drip and gravity sources). The general idea is to describe the subsurface water patterns statistically in terms of only a few (often 3) parameters which can then be predicted by the ANN. It was shown that ellipses (in two dimensions) or ellipsoids (in three dimensions) can be depicted about the center of the plume. Any fraction of water added can be related to a ‘‘probability’’ curve relating the size of the ellipse (or ellipsoid) that contains that amount of water. The initial test of an ANN to predict the moments (and hence the water plume) was with numerically generated data for infiltration from surface and subsurface drip line and point sources in three contrasting soils. The underlying dataset consisted of 1,684,500 vectors (5 soils×5 discharge rates×3 initial conditions×1,123 nodes×20 print times) where each vector had eleven elements consisting of initial water content, hydraulic properties of the soil, flow rate, time and space coordinates. The output is an estimate of subsurface water distribution for essentially any soil property, initial condition or flow rate from a drip source. Following the formal development of the ANN, we have prepared a “user-friendly” version in a spreadsheet environment (in “Excel”). The input data are selected from appropriate values and the output is instantaneous resulting in a picture of the resulting water plume. The MA has also proven valuable, on its own merit, in the description of the flow in soil under laboratory conditions for both wettable and repellant soils. This includes non-Darcian flow examples and redistribution and well as infiltration. Field experiments were conducted in different agricultural fields and various water qualities in Israel. The obtained results will be the basis for the further ANN models development. Regions of high repellence were identified primarily under the canopy of various orchard crops, including citrus and persimmons. Also, increasing OM in the applied water lead to greater repellency. Major scientific implications are that the ANN offers an alternative to conventional flow and transport modeling and that MA is a powerful technique for describing the subsurface water distributions for normal (wettable) and repellant soil. Implications of the field measurements point to the special role of OM in affecting wettability, both from the irrigation water and from soil accumulation below canopies. Implications for agriculture are that a modified approach for drip system design should be adopted for open area crops and orchards, and taking into account the OM components both in the soil and in the applied waters.