Finite Expression Method with TranNet-based Function Learning for High-Dimensional Partial Differential Equations

TL;DR

This paper introduces an extension of the finite expression method (FEX) using TranNet-based neural network operators to efficiently solve high-dimensional PDEs, overcoming the curse of dimensionality.

Contribution

The paper presents a novel FEX extension with neural network operators initialized by TransNet, improving scalability and accuracy for high-dimensional PDE solutions.

Findings

The extended method achieves high accuracy in high-dimensional PDEs.

Numerical experiments demonstrate the effectiveness of the neural network-based FEX.

The approach offers polynomial memory complexity and favorable computational costs.

Abstract

In this paper, we study a machine-learning-based solver for high-dimensional partial differential equations (PDEs). Computing accurate solutions efficiently for such problems remains challenging because of the curse of dimensionality, which severely limits the scalability of classical numerical methods. Our approach builds on the recently developed finite expression method (FEX), which approximates PDE solutions in a function space generated by finitely many analytic expressions. This framework has been shown to achieve high, and in some cases machine-level, accuracy with polynomial memory complexity and favorable computational cost. We propose an extension of FEX in which the functional pool is generated by shallow neural network operators whose parameters are initialized using the transferable neural network method TransNet. Numerical experiments suggest that the proposed extension is an effective alternative for solving several high-dimensional PDEs.