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Learning to Control PDEs with Differentiable Physics [article]

Philipp Holl, Vladlen Koltun, Nils Thuerey
2020 arXiv   pre-print
A variety of such tasks involves continuous physical systems, which can be described by partial differential equations (PDEs) with many degrees of freedom.  ...  Both stages are trained end-to-end using a differentiable PDE solver.  ...  RESULTS We evaluate the capabilities of our method to learn to control physical PDEs in three different test environments of increasing complexity.  ... 
arXiv:2001.07457v1 fatcat:z4hv6a5z6nacfnpyee4jwwlcxu

Neural Differential Equations as a Basis for Scientific Machine Learning (SciML) [article]

Christopher Rackauckas
Problems such as optimal control and automated learning of differential equation models will be reduced to training problems on neural differential equations.  ...  Scientific Machine Learning (SciML) is an emerging discipline which merges the mechanistic models of science and engineering with non-mechanistic machine learning models to solve problems which were previously  ...  Learn the unknown function via neural network.  ALLOW FOR AUTOMATICALLY LEARNING MODELS, USING KNOWN EQUATIONS AS A PRIOR  SOLVE OPTIMAL CONTROL PROBLEMS  ACCELERATE THE SOLUTION OF PDES  SOLVE PDES  ... 
doi:10.6084/m9.figshare.12751955.v1 fatcat:zhwjvt23tfhmjljetsfobsv5q4

Fast PDE-constrained optimization via self-supervised operator learning [article]

Sifan Wang, Mohamed Aziz Bhouri, Paris Perdikaris
2021 arXiv   pre-print
In this work we leverage physics-informed deep operator networks (DeepONets) -- a self-supervised framework for learning the solution operator of parametric PDEs -- to build fast and differentiable surrogates  ...  In cases where the experimental dynamics can be described by partial differential equations (PDEs), such problems can be mathematically translated into PDE-constrained optimization tasks, which quickly  ...  We would also like to thank the developers of the software that enabled our research, including JAX [54] , Matplotlib [55] , NumPy [56] , FEniCS [52] , and dolfin-adjoint [57] .  ... 
arXiv:2110.13297v1 fatcat:cnb7ewvac5cqviwaiyb5o3kd5a

Recent Advancements in Differential Equation Solver Software [article]

Christopher Rackauckas
, differential equations which incorporate trainable latent neural networks into their derivative functions to automatically learn dynamics from data.  ...  However, many applications of differential equations still rely on the same older software, possibly to their own detriment.  ...  u Once learned, the PDE solution is known.  ... 
doi:10.6084/m9.figshare.12751997.v1 fatcat:o7lbctk2evfnxha4b4wrxnt26e

Solving PDE-constrained Control Problems using Operator Learning [article]

Rakhoon Hwang, Jae Yong Lee, Jin Young Shin, Hyung Ju Hwang
2021 arXiv   pre-print
We demonstrate the successful application of our method to various optimal control problems for different control variables with diverse PDE constraints from the Poisson equation to Burgers' equation.  ...  We propose a novel framework that is generally applicable to solving PDE-constrained optimal control problems by introducing surrogate models for PDE solution operators with special regularizers.  ...  Learning to B.; Logg, A.; Richardson, C.; Ring, J.; Rognes, M. E.; and control pdes with differentiable physics. arXiv preprint Wells, G. N. 2015. The FEniCS project version 1.5.  ... 
arXiv:2111.04941v2 fatcat:3ziq3jnnsfe7nozuul67vilsii

Three Ways to Solve Partial Differential Equations with Neural Networks – A Review [article]

Jan Blechschmidt, Oliver G. Ernst
2021 arXiv   pre-print
Neural networks are increasingly used to construct numerical solution methods for partial differential equations.  ...  , methods based on the Feynman-Kac formula and methods based on the solution of backward stochastic differential equations.  ...  Other areas prone to high-dimensional PDE problems include stochastic control, differential games and quantum physics.  ... 
arXiv:2102.11802v2 fatcat:xc647il5q5f4baixs74arorbbm

Partial Differential Equations is All You Need for Generating Neural Architectures – A Theory for Physical Artificial Intelligence Systems [article]

Ping Guo, Kaizhu Huang, Zenglin Xu
2021 arXiv   pre-print
to physical artificial intelligence.  ...  The learning strategies, such as Adaptive moment estimation, L-BFGS, pseudoinverse learning algorithms and partial differential equation constrained optimization, are also presented.  ...  For example, Han et al have used deep learning to solve high-dimensional PDEs [63] , including solving Hamilton -Jacobi -Bellman equation for control.  ... 
arXiv:2103.08313v1 fatcat:pgfp2k3y3nhzlghybgf2thubxe

PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain [article]

Han Gao, Luning Sun, Jian-Xun Wang
2020 arXiv   pre-print
In this paper, we propose a novel physics-constrained CNN learning architecture, aiming to learn solutions of parametric PDEs on irregular domains without any labeled data.  ...  Recently, the advent of deep learning has spurred interest in the development of physics-informed neural networks (PINN) for efficiently solving partial differential equations (PDEs), particularly in a  ...  We also gratefully acknowledge the discussion with Dr. Matthew J. Zahr in the early stage of this research.  ... 
arXiv:2004.13145v2 fatcat:yopnmvhjsvdflmmv7ycggbhx5m

PDE-Net: Learning PDEs from Data [article]

Zichao Long, Yiping Lu, Xianzhong Ma, Bin Dong
2018 arXiv   pre-print
The basic idea of the proposed PDE-Net is to learn differential operators by learning convolution kernels (filters), and apply neural networks or other machine learning methods to approximate the unknown  ...  In this paper, we present an initial attempt to learn evolution PDEs from data.  ...  To differentiate the two cases, we shall call the PDE-Net with frozen filters "the Frozen-PDE-Net".  ... 
arXiv:1710.09668v2 fatcat:bu6gvt7bfvcfppamy5lhpwcr6m

Solving Partial Differential Equations with Bernstein Neural Networks [chapter]

Sina Razvarz, Raheleh Jafari, Alexander Gegov
2018 Advances in Intelligent Systems and Computing  
In this paper, a neural network-based procedure is suggested to produce estimated solutions (controllers) for the second-order nonlinear partial differential equations (PDEs).  ...  This concept is laid down so as to produce a prevalent approximation on the basis of the learning method which is at par with quasi-Newton rule.  ...  It is controller design process. The trial solution related to the PDE is stated as an addition of two parts.  ... 
doi:10.1007/978-3-319-97982-3_5 fatcat:3lnvpo6oabeupaoifyhrzuwxuy

Learning PDEs for Image Restoration via Optimal Control [chapter]

Risheng Liu, Zhouchen Lin, Wei Zhang, Zhixun Su
2010 Lecture Notes in Computer Science  
Partial differential equations (PDEs) have been successfully applied to many computer vision and image processing problems.  ...  of differential invariants, which is data-driven and can effectively adapt to different problems and complex conditions.  ...  We will present a PDE-based optimal control framework to learn these coefficient functions in Section 3. Table 2 .  ... 
doi:10.1007/978-3-642-15549-9_9 fatcat:q2456nvonffwhmzv5rbruthqum

Model Reduction of Swing Equations with Physics Informed PDE [article]

Laurent Pagnier, Michael Chertkov, Julian Fritzsch, Philippe Jacquod
2021 arXiv   pre-print
We finally discuss future extensions of this work, where the presented PDE-based reduced modeling will initialize a physics-informed machine learning approach for real-time modeling, n-1 feasibility assessment  ...  We suggest, following Seymlyen (1974) and Thorpe, Seyler and Phadke (1999), to map the swing equations onto a linear, inhomogeneous Partial Differential Equation (PDE) of parabolic type in two space and  ...  ACKNOWLEDGMENT The authors thank Robert Ferrando and Christopher Koh for participating in the discussions which led to the manuscript.  ... 
arXiv:2110.14066v1 fatcat:uk5vjn62qndjjpbmefowj4mimq

A Latent space solver for PDE generalization [article]

Rishikesh Ranade, Chris Hill, Haiyang He, Amir Maleki, Jay Pathak
2021 arXiv   pre-print
In this work we propose a hybrid solver to solve partial differential equation (PDE)s in the latent space.  ...  The solver uses an iterative inferencing strategy combined with solution initialization to improve generalization of PDE solutions.  ...  These approaches use differentiable solvers to learn and control PDE solutions as well as the dynamics of the system (Amos & Kolter, 2017; Um et al., 2020; de Avila Belbute-Peres et al., 2018; Toussaint  ... 
arXiv:2104.02452v1 fatcat:uz6i7572dfb67paxcyh3zo4pka

The Dawning of a New Era in Applied Mathematics

Weinan E
2021 Notices of the American Mathematical Society  
These algorithms have opened the door to dealing with real-world control problems and high-dimensional PDEs.  ...  To put it simply, the former is about physics and the latter is about differential equations.  ...  The new start makes linear algebra more approachable to students with a wide range of backgrounds.  ... 
doi:10.1090/noti2259 fatcat:zzr4wbe5tfcsdoljqr7gluytby

Adversarial Multi-task Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations [article]

Pongpisit Thanasutives, Masayuki Numao, Ken-ichi Fukui
2021 arXiv   pre-print
The multi-task scheme exploits the benefits of learning shared representations, controlled by cross-stitch modules, between multiple related PDEs, which are obtainable by varying the PDE parameterization  ...  Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation.  ...  Solving PDEs using neural networks plays an essential role in the development of a hybrid system between physics and machine learning, owing to the physically consistent network outputs as the PDE solutions  ... 
arXiv:2104.14320v2 fatcat:gfd4hquc3batlmpprm44du47fi
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