Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Refined generalization analysis of the Deep Ritz Method and Physics-Informed Neural Networks
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Derives localized (fast-rate) generalization bounds for Deep Ritz Method on Poisson and static Schrödinger equations with Neumann BC, and for PINNs on linear second-order elliptic PDEs via a multi-task Talagrand inequality, including Barron and Sobolev function-class settings.
Problem
Generalization error of neural-network PDE solvers (Deep Ritz Method and PINNs) lacks sharp, dimension-explicit rates; prior bounds were slow-rate, required strong convexity or H_0^1 function classes, or only covered special PDEs.
Approach
Uses local Rademacher complexity and a new error decomposition (strong convexity of energy functional) for DRM with Neumann BC, approximates Barron functions in H^1 by two-layer ReLU nets with controlled weights, and extends the entropy method to a multi-task Talagrand inequality so PINN losses can be analyzed as MTL without strong convexity.
Key results
- For the DRM Poisson and Schrödinger problems with Barron solutions, the excess energy risk is bounded by (d/n)^((3d+2)/(2(3d+1))) log n up to constants, improving on the n^(-1/3) rate of prior work.
- For Sobolev solutions in W^{k,inf}, the DRM bound improves the rate from n^(-(2k-2)/(d+4k-4)) to n^(-(2k-2)/(d+2k-2)) (log factors aside).
- For PINNs on linear second-order elliptic PDEs, the loss bound for Barron solutions in B^3 is O(m log n / n + m^(-(1+2/(3d)))+t/n), and for Sobolev solutions the rate is n^(-(2k-4)/(d+2k-4)) log n.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use localization analysis for the Deep Ritz Method | Deep Ritz Method on the static Schrödinger equation · Static Schrödinger equation, Sobolev solution | energy excess rate: n^{-\frac{2k-2}{d+4k-4}} -> n^{-\frac{2k-2}{d+2k-2}} | Helped |
| Use localized multi-task learning analysis without requiring strong convexity | PINNs for linear second-order elliptic PDEs · Linear second-order elliptic PDE, Sobolev solution | PINN loss rate: n^{-\frac{2k-4}{d+4k-8}} -> n^{-\frac{2k-4}{d+2k-4}} | Helped |
| Use the entropy-method Talagrand-type concentration inequality | Multi-task learning analysis for PINNs · Multi-task concentration bound | bound constants | Helped |
| Use localization for the DRM Barron-solution analysis | Deep Ritz Method · Poisson and static Schrödinger equations, Barron solutions | generalization error rate: n^{-\frac{1}{3}} -> (\frac{d}{n})^{\frac{3d+2}{2(3d+1)}} | Helped |
Limitations
- Results are theoretical; no numerical experiments validate the bounds.
- For PINNs, the boundary term needed for H^2 error control (H^{3/2} norm of g) is hard to handle in the covering number analysis and is left to future work.
- XPINNs and other PINN variants do not fit the standard MTL framework and are not covered.
Experimental setup
Metrics: generalization error bound (excess energy risk) H^1 error bound convergence rate in n
Baselines: Duan et al. DRM generalization bound Lu et al. PINN generalization bound Mishra and Molinaro PINN bound
Builds on: Deep Ritz Method Physics-Informed Neural Networks local Rademacher complexity Talagrand-type inequality for multi-task learning
Closest papers by meaning
- Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality (ICLR 2022 · similarity 0.95)
- Learning from Integral Losses in Physics Informed Neural Networks (ICML 2024 · similarity 0.87)
- Number Theoretic Accelerated Learning of Physics-Informed Neural Networks (AAAI 2025 · similarity 0.86)
- Global convergence of adjoint-optimized neural PDEs (NeurIPS 2026 · similarity 0.86)
- Optimal deep learning of holomorphic operators between Banach spaces (NeurIPS 2024 · similarity 0.86)
- Parameterized Physics-informed Neural Networks for Parameterized PDEs (ICML 2024 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.