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A Priori Generalization Analysis of the Deep Ritz Method for Solving High Dimensional Elliptic Equations

arXiv 2101.01708 ↗First posted 5 Jan 2021math.NA cs.LG cs.NAcited by 18

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Abstract (arXiv)

This paper concerns the a priori generalization analysis of the Deep Ritz Method (DRM) [W. E and B. Yu, 2017], a popular neural-network-based method for solving high dimensional partial differential equations. We derive the generalization error bounds of two-layer neural networks in the framework of the DRM for solving two prototype elliptic PDEs: Poisson equation and static Schr"odinger equation on the d-dimensional unit hypercube. Specifically, we prove that the convergence rates of generalization errors are independent of the dimension d, under the a priori assumption that the exact solutions of the PDEs lie in a suitable low-complexity space called spectral Barron space. Moreover, we give sufficient conditions on the forcing term and the potential function which guarantee that the solutions are spectral Barron functions. We achieve this by developing a new solution theory for the PDEs on the spectral Barron space, which can be viewed as an analog of the classical Sobolev regularity theory for PDEs.

Key facts. Cited by 18 papers in the map (data through 14 Oct 2026). Builds on 20 of its 48 references that are in the map. First posted on arXiv 5 Jan 2021.

Builds on

20 of 48 references.

References · 20

  1. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,621
  2. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  3. 2018On Lazy Training in Differentiable Programmingcited by 675
  4. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  5. 2016Error bounds for approximations with deep ReLU networkscited by 574
  6. 2017Solving high-dimensional partial differential equations using deep learningcited by 556
  7. 2018A Mean Field View of the Landscape of Two-Layers Neural Networkscited by 466
  8. 2018On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transportcited by 426
  9. 2018Optimal approximation of continuous functions by very deep ReLU networkscited by 166
  10. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 116
  11. 2020Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don'tcited by 95
  12. 2019Limitations of Lazy Training of Two-layers Neural Networkscited by 87
8 more
  1. 2020A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networkscited by 87
  2. 2016Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with ℓ1 and ℓ0 Controlscited by 84
  3. 2018A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationscited by 55
  4. 2018Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equationscited by 45
  5. 2020Two-Layer Neural Networks for Partial Differential Equations: Optimization and Generalization Theorycited by 41
  6. 2019The Barron Space and the Flow-induced Function Spaces for Neural Network Modelscited by 39
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  8. 2018Solving for high dimensional committor functions using artificial neural networkscited by 34

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 8; 2022: 5; 2023: 3; 2024: 1; 2025: 0; 2026: 1.

18 papers.

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  1. 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 79
  2. 2021DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equationscited by 66
  3. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 58
  4. 2022Deep Nonparametric Estimation of Operators between Infinite Dimensional Spacescited by 45
  5. 2021On the Representation of Solutions to Elliptic PDEs in Barron Spacescited by 25
  6. 2021A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problemscited by 19
  7. 2022IAE-Net: Integral Autoencoders for Discretization-Invariant Learningcited by 15
  8. 2022Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspectivecited by 15
  9. 2021Error Estimates for the Deep Ritz Method with Boundary Penaltycited by 14
  10. 2021Solving PDEs on Unknown Manifolds with Machine Learningcited by 12
  11. 2022Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descentcited by 11
  12. 2021Stationary Density Estimation of Ito Diffusions Using Deep Learningcited by 9
6 more
  1. 2023Convergence Analysis of the Deep Galerkin Method for Weak Solutionscited by 4
  2. 2021Generalization Error Analysis of Neural networks with Gradient Based Regularizationcited by 3
  3. 2022Convergence analysis of a quasi-Monte Carlo-based deep learning algorithm for solving partial differential equationscited by 1
  4. 2023Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Modelscited by 1
  5. 2024Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Biascited by 0
  6. 2026Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solverscited by 0