Solving Partial Differential Equations with Random Feature Models

TL;DR

This paper introduces a random feature-based framework for efficiently solving partial differential equations, offering a computationally simpler alternative to neural network-based methods with theoretical guarantees.

Contribution

It presents a novel random feature method for PDEs, providing error analysis and demonstrating advantages over existing neural network approaches.

Findings

Reduces computational complexity for PDE solving

Provides theoretical error bounds for the method

Achieves competitive results on benchmark PDEs

Abstract

Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and kernel method. In this paper, we introduce a random feature based framework toward efficiently solving PDEs. Random feature method was originally proposed to approximate large-scale kernel machines and can be viewed as a shallow neural network as well. We provide an error analysis for our proposed method along with comprehensive numerical results on several PDE benchmarks. In contrast to the state-of-the-art solvers that face challenges with a large number of collocation points, our proposed method reduces the computational complexity. Moreover, the implementation of our method is simple and does not require additional computational resources. Due to the theoretical guarantee and advantages in computation, our approach is proven to be efficient for solving PDEs.