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A Priori Estimation of the Approximation, Optimization and Generalization Errors of Random Neural Networks for Solving Partial Differential Equations

Xianliang Xu, Ye Li, Zhongyi Huang

TL;DR

The work tackles solving PDEs with PINNs by employing two-layer random neural networks whose hidden weights are sampled from explicit priors, focusing on Barron-type target functions and their Sobolev approximations. It derives rigorous approximation bounds in $H^2$ for Barron functions, and develops optimization and generalization analyses within the PINN framework, showing that projected gradient descent can achieve fast rates when strong convexity holds, with ridge regression offering an alternative in unbounded-weight scenarios. Theoretical results are complemented by experiments, including a 1D Poisson test confirming $\mathcal{O}(1/m)$ decay and a Reaction-Diffusion problem achieving high accuracy with substantial speedups over standard PINNs. The study highlights the potential and limitations of random-feature PDE solvers, noting dimensionality constraints and suggesting extensions to broader PDE solvers and function classes for future work.

Abstract

In recent years, neural networks have achieved remarkable progress in various fields and have also drawn much attention in applying them on scientific problems. A line of methods involving neural networks for solving partial differential equations (PDEs), such as Physics-Informed Neural Networks (PINNs) and the Deep Ritz Method (DRM), has emerged. Although these methods outperform classical numerical methods in certain cases, the optimization problems involving neural networks are typically non-convex and non-smooth, which can result in unsatisfactory solutions for PDEs. In contrast to deterministic neural networks, the hidden weights of random neural networks are sampled from some prior distribution and only the output weights participate in training. This makes training much simpler, but it remains unclear how to select the prior distribution. In this paper, we focus on Barron type functions and approximate them under Sobolev norms by random neural networks with clear prior distribution. In addition to the approximation error, we also derive bounds for the optimization and generalization errors of random neural networks for solving PDEs when the solutions are Barron type functions.

A Priori Estimation of the Approximation, Optimization and Generalization Errors of Random Neural Networks for Solving Partial Differential Equations

TL;DR

The work tackles solving PDEs with PINNs by employing two-layer random neural networks whose hidden weights are sampled from explicit priors, focusing on Barron-type target functions and their Sobolev approximations. It derives rigorous approximation bounds in for Barron functions, and develops optimization and generalization analyses within the PINN framework, showing that projected gradient descent can achieve fast rates when strong convexity holds, with ridge regression offering an alternative in unbounded-weight scenarios. Theoretical results are complemented by experiments, including a 1D Poisson test confirming decay and a Reaction-Diffusion problem achieving high accuracy with substantial speedups over standard PINNs. The study highlights the potential and limitations of random-feature PDE solvers, noting dimensionality constraints and suggesting extensions to broader PDE solvers and function classes for future work.

Abstract

In recent years, neural networks have achieved remarkable progress in various fields and have also drawn much attention in applying them on scientific problems. A line of methods involving neural networks for solving partial differential equations (PDEs), such as Physics-Informed Neural Networks (PINNs) and the Deep Ritz Method (DRM), has emerged. Although these methods outperform classical numerical methods in certain cases, the optimization problems involving neural networks are typically non-convex and non-smooth, which can result in unsatisfactory solutions for PDEs. In contrast to deterministic neural networks, the hidden weights of random neural networks are sampled from some prior distribution and only the output weights participate in training. This makes training much simpler, but it remains unclear how to select the prior distribution. In this paper, we focus on Barron type functions and approximate them under Sobolev norms by random neural networks with clear prior distribution. In addition to the approximation error, we also derive bounds for the optimization and generalization errors of random neural networks for solving PDEs when the solutions are Barron type functions.
Paper Structure (19 sections, 4 theorems, 131 equations, 2 figures)

This paper contains 19 sections, 4 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related Works
  4. Random Neural Networks in Machine Learning.
  5. Random Neural Networks in Scientific Computing.
  6. Notations
  7. Preliminaries
  8. Main Results
  9. Approximation Results
  10. Proof Sketch:
  11. Optimization and Generalization Results
  12. Experimental Results
  13. Verification of Theorem
  14. Application
  15. Conclusion and Discussion
  16. ...and 4 more sections

Key Result

Theorem 1

Let $P_1$ be the uniform distribution of the domain $\{\omega\in\mathbb{R}^d: \|\omega\|_1\leq M\}$ and $P_2$ be the uniform distribution of the domain $\{b\in\mathbb{R}: |b|\leq 2M\}$ with a constant $M\geq2$. Let $W_1,\cdots, W_m\sim P_1$ and $B_1,\cdots, B_m\sim P_2$, then there exists a $\mathbb and for $s\geq 6$, where and $\lesssim$ indicates that a universal multiplicative constant is om

Figures (2)

  • Figure 1: Testing loss for the Poisson equation. A random neural network with $m$ hidden nodes ($m$ ranges from 50 to 500) is used to solve 1D Poisson equation with PINN method. The dots show the estimated loss $L(m)$, the line shows the decay implied by the theoretical results $\frac{50L(50)}{m}$.
  • Figure 2: Comparison between the reference and predicted solutions for the Reaction-Diffusion equation.

Theorems & Definitions (16)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Remark 4
  • Remark 5
  • Theorem 3
  • Remark 6
  • Remark 7
  • ...and 6 more