mlsanityData to 14 Oct 2026

Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality

arXiv 2110.06897 ↗First posted 13 Oct 2021math.NA cs.LG cs.NAcited by 40

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Abstract (arXiv)

In this paper, we study the statistical limits of deep learning techniques for solving elliptic partial differential equations (PDEs) from random samples using the Deep Ritz Method (DRM) and Physics-Informed Neural Networks (PINNs). To simplify the problem, we focus on a prototype elliptic PDE: the Schr"odinger equation on a hypercube with zero Dirichlet boundary condition, which has wide application in the quantum-mechanical systems. We establish upper and lower bounds for both methods, which improves upon concurrently developed upper bounds for this problem via a fast rate generalization bound. We discover that the current Deep Ritz Methods is sub-optimal and propose a modified version of it. We also prove that PINN and the modified version of DRM can achieve minimax optimal bounds over Sobolev spaces. Empirically, following recent work which has shown that the deep model accuracy will improve with growing training sets according to a power law, we supply computational experiments to show a similar behavior of dimension dependent power law for deep PDE solvers.

Key facts. Cited by 40 papers in the map (data through 14 Oct 2026). First posted on arXiv 13 Oct 2021.

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2022: 7; 2023: 12; 2024: 10; 2025: 3; 2026: 8.

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  1. 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 92
  2. 2024Numerical analysis of physics-informed neural networks and related models in physics-informed machine learningcited by 39
  3. 2022Is L2 Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?cited by 23
  4. 2023Distribution learning via neural differential equations: a nonparametric statistical perspectivecited by 22
  5. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  6. 2022Minimax Optimal Kernel Operator Learning via Multilevel Trainingcited by 13
  7. 2023Minimax Instrumental Variable Regression and L2 Convergence Guarantees without Identification or Closednesscited by 13
  8. 2024On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networkscited by 13
  9. 2022Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descentcited by 11
  10. 2022Born-Infeld (BI) for AI: Energy-Conserving Descent (ECD) for Optimizationcited by 10
  11. 2023Source Condition Double Robust Inference on Functionals of Inverse Problemscited by 9
  12. 2024On Bellman equations for continuous-time policy evaluation I: discretization and approximationcited by 7
28 more
  1. 2024UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEscited by 6
  2. 2022Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Methodcited by 4
  3. 2023LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometrycited by 4
  4. 2023Convergence Analysis of the Deep Galerkin Method for Weak Solutionscited by 4
  5. 2023Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networkscited by 3
  6. 2023Solving Elliptic Optimal Control Problems via Neural Networks and Optimality Systemcited by 3
  7. 2024On the estimation rate of Bayesian PINN for inverse problemscited by 3
  8. 2024Solving High-Dimensional Partial Integral Differential Equations: The Finite Expression Methodcited by 3
  9. 2025Is RL fine-tuning harder than regression? A PDE learning approach for diffusion modelscited by 3
  10. 2023Statistical Spatially Inhomogeneous Diffusion Inferencecited by 2
  11. 2025Unifying Learning Dynamics and Generalization in Transformers Scaling Lawcited by 2
  12. 2022Convergence analysis of a quasi-Monte Carlo-based deep learning algorithm for solving partial differential equationscited by 1
  13. 2023Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Modelscited by 1
  14. 2023Score Operator Newton transportcited by 1
  15. 2023Learning Green's Function Efficiently Using Low-Rank Approximationscited by 1
  16. 2024Error Analysis of Three-Layer Neural Network Trained with PGD for Deep Ritz Methodcited by 1
  17. 2024Point Source Identification Using Singularity Enriched Neural Networkscited by 1
  18. 2026Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problemscited by 1
  19. 2026Elliptic Regularity Theory in Barron Spaces and Applications to the Deep Ritz Methodcited by 1
  20. 2024Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Biascited by 0
  21. 2024DNA-SE: Towards Deep Neural-Nets Assisted Semiparametric Estimationcited by 0
  22. 2025Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domainscited by 0
  23. 2026Continuous-time reinforcement learning: ellipticity enables model-free value function approximationcited by 0
  24. 2026Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEscited by 0
  25. 2026Breakeven complexity: A new perspective on neural partial differential equation solverscited by 0
  26. 2026A PAC-Bayesian View of Generalisation for Physics-Informed Machine Learningcited by 0
  27. 2026Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Methodcited by 0
  28. 2026Sobolev Regularized Score Difference Estimation in Diffusion Modelscited by 0