mlsanityData to 7 Oct 2026

Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems

arXiv 2212.03130 ↗First posted 6 Dec 2022cs.LG cs.CE cs.NAcited by 34

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Abstract (arXiv)

Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where deep learning algorithms are used to solve problems in mathematics. The latter has popularised the field of scientific machine learning where deep learning is applied to problems in scientific computing. Specifically, more and more neural network architectures have been developed to solve specific classes of partial differential equations (PDEs). Such methods exploit properties that are inherent to PDEs and thus solve the PDEs better than standard feed-forward neural networks, recurrent neural networks, or convolutional neural networks. This has had a great impact in the area of mathematical modeling where parametric PDEs are widely used to model most natural and physical processes arising in science and engineering. In this work, we review such methods as well as their extensions for parametric studies and for solving the related inverse problems. We equally proceed to show their relevance in some industrial applications.

Key facts. Cited by 34 papers in the map (data through 7 Oct 2026). Builds on 87 of its 177 references that are in the map. First posted on arXiv 6 Dec 2022.

Builds on

87 of 177 references.

References · 87

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2016Gaussian Error Linear Units (GELUs)cited by 3,431
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  4. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,597
  5. 2015Automatic differentiation in machine learning: a surveycited by 846
  6. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  7. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  8. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 629
  9. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  10. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  11. 2017Self-Normalizing Neural Networkscited by 591
  12. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
38 more
  1. 2016Error bounds for approximations with deep ReLU networkscited by 567
  2. 2017Solving high-dimensional partial differential equations using deep learningcited by 549
  3. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  4. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  5. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  6. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  7. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  8. 2017Stable Architectures for Deep Neural Networkscited by 340
  9. 2021U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flowcited by 292
  10. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  11. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  12. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  13. 2017Learned Primal-dual Reconstructioncited by 251
  14. 2018Analyzing Inverse Problems with Invertible Neural Networkscited by 239
  15. 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 233
  16. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  17. 2021Gradient-enhanced physics-informed neural networks for forward and inverse PDE problemscited by 220
  18. 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 203
  19. 2018Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantificationcited by 199
  20. 2018Deep Neural Networks Motivated by Partial Differential Equationscited by 187
  21. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
  22. 2017Optimal approximation of piecewise smooth functions using deep ReLU neural networkscited by 176
  23. 2018Universality of Deep Convolutional Neural Networkscited by 176
  24. 2020Deep Network Approximation for Smooth Functionscited by 163
  25. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  26. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  27. 2022Physics-Informed Deep Neural Operator Networkscited by 110
  28. 2020A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEscited by 105
  29. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  30. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  31. 2019Error bounds for approximations with deep ReLU neural networks in Ws,p normscited by 95
  32. 2017Optimal Approximation with Sparsely Connected Deep Neural Networkscited by 90
  33. 2018NETT: Solving Inverse Problems with Deep Neural Networkscited by 89
  34. 2021Improved architectures and training algorithms for deep operator networkscited by 85
  35. 2021POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decompositioncited by 79
  36. 2020Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learningcited by 77
  37. 2022Wavelet neural operator: a neural operator for parametric partial differential equationscited by 77
  38. 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74

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Citing papers per year: 2022: 1; 2023: 8; 2024: 9; 2025: 12; 2026: 4.

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  3. 2023Deep Learning in Deterministic Computational Mechanicscited by 21
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  6. 2024Exact Enforcement of Temporal Continuity in Sequential Physics-Informed Neural Networkscited by 9
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  8. 2024An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learningcited by 6
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  10. 2023A Direct Sampling-Based Deep Learning Approach for Inverse Medium Scattering Problemscited by 4
  11. 2023Inverse Problems with Learned Forward Operatorscited by 4
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  6. 2023Electrical Impedance Tomography: A Fair Comparative Study on Deep Learning and Analytic-based Approachescited by 2
  7. 2024GN-SINDy: Greedy Sampling Neural Network in Sparse Identification of Nonlinear Partial Differential Equationscited by 2
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  9. 2025Convergence of Stochastic Gradient Methods for Wide Two-Layer Physics-Informed Neural Networks for the Poisson Equationcited by 2
  10. 2026Autoregression-Free Neural Operators for Time-Dependent PDEscited by 2
  11. 2025An Iterative Deep Ritz Method for Monotone Elliptic Problemscited by 1
  12. 2025Solved in Unit Domain: JacobiNet for Differentiable Coordinate-Transformed PINNscited by 1
  13. 2025Noise-adapted Neural Operator for Robust Non-Line-of-Sight Imagingcited by 1
  14. 2025Physically consistent model learning for reaction-diffusion systemscited by 1
  15. 2026Understanding the Theoretical Foundations of Deep Neural Networks through Differential Equationscited by 1
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  18. 2025Estimating properties of a homogeneous bounded soil using machine learning modelscited by 0
  19. 2025Discretization-independent operator learning for partial differential equationscited by 0
  20. 2025Benchmarking Physics-Informed Neural Networks and Boundary Elements Methods for Wave Scatteringcited by 0
  21. 2026Symbolic recovery of PDEs from measurement datacited by 0
  22. 2026The Differential Neural Tangent Kernel and Its Positivitycited by 0