mlsanityData to 7 Oct 2026

An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning

arXiv 2408.13222 ↗First posted 23 Aug 2024math.NA cs.NA stat.MLcited by 6

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Abstract (arXiv)

The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively studied. One class of methods which has received a lot of attention in recent years are machine learning-based methods, which typically involve the training of artificial neural networks (ANNs) by means of stochastic gradient descent type optimization methods. While approximation methods for PDEs using ANNs have first been proposed in the 1990s they have only gained wide popularity in the last decade with the rise of deep learning. This article aims to provide an introduction to some of these methods and the mathematical theory on which they are based. We discuss methods such as physics-informed neural networks (PINNs) and deep BSDE methods and consider several operator learning approaches.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 33 of its 94 references that are in the map. First posted on arXiv 23 Aug 2024.

Builds on

33 of 94 references.

References · 33

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
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  4. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
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21 more
  1. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  2. 2023Learning stiff chemical kinetics using extended deep neural operatorscited by 30
  3. 2022Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuitiescited by 29
  4. 2023Koopman neural operator as a mesh-free solver of non-linear partial differential equationscited by 29
  5. 2021Neural networks-based algorithms for stochastic control and PDEs in financecited by 26
  6. 2022Partial Differential Equations Meet Deep Neural Networks: A Surveycited by 25
  7. 2023Deep Operator Network Approximation Rates for Lipschitz Operatorscited by 24
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  16. 2022Enhanced DeepONet for Modeling Partial Differential Operators Considering Multiple Input Functionscited by 11
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  21. 2023Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equationscited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 2; 2025: 2; 2026: 2.

6 papers.

  1. 2024Optimal Convergence Rates for Neural Operatorscited by 7
  2. 2024Deep Operator BSDE: a Numerical Scheme to Approximate Solution Operatorscited by 0
  3. 2025Error Analysis of Deep PDE Solvers for Option Pricingcited by 0
  4. 2025Error analysis for the deep Kolmogorov methodcited by 0
  5. 2026Random Neural Network Expressivity for Non-Linear Partial Differential Equationscited by 0
  6. 2026Gradient-Update Mismatch: Rethinking Conflict-Free Training of Physics-Informed Neural Networkscited by 0