mlsanityData to 7 Oct 2026

Neural Control of Parametric Solutions for High-dimensional Evolution PDEs

arXiv 2302.00045 ↗First posted 31 Jan 2023math.NA cs.LG cs.NAcited by 5

Not on the map. Why?

Abstract (arXiv)

We develop a novel computational framework to approximate solution operators of evolution partial differential equations (PDEs). By employing a general nonlinear reduced-order model, such as a deep neural network, to approximate the solution of a given PDE, we realize that the evolution of the model parameter is a control problem in the parameter space. Based on this observation, we propose to approximate the solution operator of the PDE by learning the control vector field in the parameter space. From any initial value, this control field can steer the parameter to generate a trajectory such that the corresponding reduced-order model solves the PDE. This allows for substantially reduced computational cost to solve the evolution PDE with arbitrary initial conditions. We also develop comprehensive error analysis for the proposed method when solving a large class of semilinear parabolic PDEs. Numerical experiments on different high-dimensional evolution PDEs with various initial conditions demonstrate the promising results of the proposed method.

Key facts. Cited by 5 papers in the map (data through 7 Oct 2026). Builds on 49 of its 95 references that are in the map. First posted on arXiv 31 Jan 2023.

Builds on

49 of 95 references.

References · 49

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  3. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  4. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  5. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  6. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  7. 2016Error bounds for approximations with deep ReLU networkscited by 567
  8. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  9. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  10. 2017Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equationscited by 362
  11. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
  12. 2020hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decompositioncited by 236
37 more
  1. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  2. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 188
  3. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 176
  4. 2015Training Very Deep Networkscited by 172
  5. 2022Respecting causality is all you need for training physics-informed neural networkscited by 154
  6. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 139
  7. 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 133
  8. 2017A unified deep artificial neural network approach to partial differential equations in complex geometriescited by 122
  9. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  10. 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
  11. 2020A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networkscited by 86
  12. 2016Why Deep Neural Networks for Function Approximation?cited by 82
  13. 2020Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path spacecited by 80
  14. 2020Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domainscited by 78
  15. 2017Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equationscited by 62
  16. 2022Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equationscited by 55
  17. 2021Evolutional Deep Neural Networkcited by 48
  18. 2021Data-driven discovery of Green's functions with human-understandable deep learningcited by 43
  19. 2020Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theorycited by 41
  20. 2020Physics-Informed Neural Networks for Nonhomogeneous Material Identification in Elasticity Imagingcited by 40
  21. 2020nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applicationscited by 36
  22. 2020Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approachcited by 35
  23. 2019A Phase Shift Deep Neural Network for High Frequency Approximation and Wave Problemscited by 34
  24. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  25. 2017Deep Learning with Domain Adaptation for Accelerated Projection-Reconstruction MRcited by 31
  26. 2020On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEscited by 31
  27. 2020Numerical Solution of Inverse Problems by Weak Adversarial Networkscited by 29
  28. 2022Learning Green's functions associated with time-dependent partial differential equationscited by 25
  29. 2019Multi-scale Deep Neural Networks for Solving High Dimensional PDEscited by 24
  30. 2018A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equationscited by 23
  31. 2022Is L2 Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?cited by 23
  32. 2022Finite Expression Method for Solving High-Dimensional Partial Differential Equationscited by 21
  33. 2018BCR-Net: a neural network based on the nonstandard wavelet formcited by 18
  34. 2022Learning the Solution Operator of Boundary Value Problems using Graph Neural Networkscited by 17
  35. 2018Neural Networks Trained to Solve Differential Equations Learn General Representationscited by 16
  36. 2021Learning Green's Functions of Linear Reaction-Diffusion Equations with Application to Fast Numerical Solvercited by 15
  37. 2022BI-GreenNet: Learning Green's functions by boundary integral networkcited by 13

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 3; 2025: 1; 2026: 1.

5 papers.

  1. 2024PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representationscited by 11
  2. 2024A Natural Primal-Dual Hybrid Gradient Method for Adversarial Neural Network Training on Solving Partial Differential Equationscited by 2
  3. 2024Systematic construction of continuous-time neural networks for linear dynamical systemscited by 1
  4. 2025HEATNETs: Explainable Random Feature Neural Networks for High-Dimensional Parabolic PDEscited by 1
  5. 2026A transition-density-based operator learning method for Fokker-Planck equations with various initial conditionscited by 0