mlsanityData to 13 Oct 2026

Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theory

arXiv 2006.15733 ↗First posted 28 Jun 2020math.NA cs.LG cs.NAcited by 41

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Abstract (arXiv)

The problem of solving partial differential equations (PDEs) can be formulated into a least-squares minimization problem, where neural networks are used to parametrize PDE solutions. A global minimizer corresponds to a neural network that solves the given PDE. In this paper, we show that the gradient descent method can identify a global minimizer of the least-squares optimization for solving second-order linear PDEs with two-layer neural networks under the assumption of over-parametrization. We also analyze the generalization error of the least-squares optimization for second-order linear PDEs and two-layer neural networks, when the right-hand-side function of the PDE is in a Barron-type space and the least-squares optimization is regularized with a Barron-type norm, without the over-parametrization assumption.

Key facts. Cited by 41 papers in the map (data through 13 Oct 2026). First posted on arXiv 28 Jun 2020.

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Citing papers per year: 2021: 13; 2022: 8; 2023: 2; 2024: 4; 2025: 9; 2026: 5.

41 papers.

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  1. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 78
  2. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 57
  3. 2021Evolutional Deep Neural Networkcited by 50
  4. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  5. 2022Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodologycited by 44
  6. 2021Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionalitycited by 34
  7. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  8. 2021Reproducing Activation Function for Deep Learningcited by 19
  9. 2021A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problemscited by 19
  10. 2021A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equationscited by 18
  11. 2021Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 16
  12. 2022IAE-Net: Integral Autoencoders for Discretization-Invariant Learningcited by 15
29 more
  1. 2021Error Estimates for the Deep Ritz Method with Boundary Penaltycited by 14
  2. 2021The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimationcited by 14
  3. 2021Solving PDEs on Unknown Manifolds with Machine Learningcited by 12
  4. 2022Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descentcited by 11
  5. 2021Stationary Density Estimation of Ito Diffusions Using Deep Learningcited by 9
  6. 2022Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networkscited by 8
  7. 2022Approximation of Functionals by Neural Network without Curse of Dimensionalitycited by 8
  8. 2022Imaging Conductivity from Current Density Magnitude using Neural Networkscited by 6
  9. 2023Neural Control of Parametric Solutions for High-dimensional Evolution PDEscited by 5
  10. 2024Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networkscited by 5
  11. 2022Convergence analysis of unsupervised Legendre-Galerkin neural networks for linear second-order elliptic PDEscited by 3
  12. 2024Convergence of Implicit Gradient Descent for Training Two-Layer Physics-Informed Neural Networkscited by 3
  13. 2025Solving Nonlinear PDEs with Sparse Radial Basis Function Networkscited by 3
  14. 2025Learn Singularly Perturbed Solutions via Homotopy Dynamicscited by 2
  15. 2025Deep Learning Optimization Using Self-Adaptive Weighted Auxiliary Variablescited by 2
  16. 2025Convergence of Stochastic Gradient Methods for Wide Two-Layer Physics-Informed Neural Networks for the Poisson Equationcited by 2
  17. 2025Curse of Dimensionality in Neural Network Optimizationcited by 1
  18. 2025Layer Separation Deep Learning Model with Auxiliary Variables for Partial Differential Equationscited by 1
  19. 2021Approximating Probability Distributions by using Wasserstein Generative Adversarial Networkscited by 0
  20. 2024Approximation of Solution Operators for High-dimensional PDEscited by 0
  21. 2024Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Biascited by 0
  22. 2025Generalizability of Neural Networks Minimizing Empirical Risk Based on Expressive Abilitycited by 0
  23. 2025Error analysis for the deep Kolmogorov methodcited by 0
  24. 2025Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domainscited by 0
  25. 2026Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solverscited by 0
  26. 2026Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Methodcited by 0
  27. 2026The Differential Neural Tangent Kernel and Its Positivitycited by 0
  28. 2026Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEscited by 0
  29. 2026Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximationcited by 0