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

arXiv: https://arxiv.org/abs/2110.06897 · HTML: https://mlsanity.com/paper/2110.06897/

## Key facts

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

## Built on by (most cited first, 30 of 40)

- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning](https://mlsanity.com/paper/2402.10926/) (2024)
- [Is L2 Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?](https://mlsanity.com/paper/2206.02016/) (2022)
- [Distribution learning via neural differential equations: a nonparametric statistical perspective](https://mlsanity.com/paper/2309.01043/) (2023)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [Minimax Optimal Kernel Operator Learning via Multilevel Training](https://mlsanity.com/paper/2209.14430/) (2022)
- [Minimax Instrumental Variable Regression and L2 Convergence Guarantees without Identification or Closedness](https://mlsanity.com/paper/2302.05404/) (2023)
- [On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networks](https://mlsanity.com/paper/2409.00901/) (2024)
- [Sobolev Acceleration and Statistical Optimality for Learning Elliptic Equations via Gradient Descent](https://mlsanity.com/paper/2205.07331/) (2022)
- [Born-Infeld (BI) for AI: Energy-Conserving Descent (ECD) for Optimization](https://mlsanity.com/paper/2201.11137/) (2022)
- [Source Condition Double Robust Inference on Functionals of Inverse Problems](https://mlsanity.com/paper/2307.13793/) (2023)
- [On Bellman equations for continuous-time policy evaluation I: discretization and approximation](https://mlsanity.com/paper/2407.05966/) (2024)
- [UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs](https://mlsanity.com/paper/2408.04846/) (2024)
- [Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method](https://mlsanity.com/paper/2209.02931/) (2022)
- [LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometry](https://mlsanity.com/paper/2302.01518/) (2023)
- [Convergence Analysis of the Deep Galerkin Method for Weak Solutions](https://mlsanity.com/paper/2302.02405/) (2023)
- [Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks](https://mlsanity.com/paper/2308.09605/) (2023)
- [Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System](https://mlsanity.com/paper/2308.11925/) (2023)
- [On the estimation rate of Bayesian PINN for inverse problems](https://mlsanity.com/paper/2406.14808/) (2024)
- [Solving High-Dimensional Partial Integral Differential Equations: The Finite Expression Method](https://mlsanity.com/paper/2410.00835/) (2024)
- [Is RL fine-tuning harder than regression? A PDE learning approach for diffusion models](https://mlsanity.com/paper/2509.02528/) (2025)
- [Statistical Spatially Inhomogeneous Diffusion Inference](https://mlsanity.com/paper/2312.05793/) (2023)
- [Unifying Learning Dynamics and Generalization in Transformers Scaling Law](https://mlsanity.com/paper/2512.22088/) (2025)
- [Convergence analysis of a quasi-Monte Carlo-based deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/2210.16196/) (2022)
- [Efficient Sampling of Stochastic Differential Equations with Positive Semi-Definite Models](https://mlsanity.com/paper/2303.17109/) (2023)
- [Score Operator Newton transport](https://mlsanity.com/paper/2305.09792/) (2023)
- [Learning Green's Function Efficiently Using Low-Rank Approximations](https://mlsanity.com/paper/2308.00350/) (2023)
- [Error Analysis of Three-Layer Neural Network Trained with PGD for Deep Ritz Method](https://mlsanity.com/paper/2405.11451/) (2024)
- [Point Source Identification Using Singularity Enriched Neural Networks](https://mlsanity.com/paper/2408.09143/) (2024)
- [Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems](https://mlsanity.com/paper/2605.04126/) (2026)

## 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.