A Least-Squares-Based Neural Network (LS-Net) for Solving Linear Parametric PDEs

TL;DR

This paper introduces LS-Net, a neural network method that combines deep learning and least-squares optimization to efficiently solve linear parametric PDEs, with theoretical guarantees and demonstrated numerical effectiveness.

Contribution

The paper presents a novel LS-Net approach that integrates neural networks with least-squares solvers for parametric PDEs, including theoretical approximation guarantees.

Findings

Successfully solves 1D and 2D parametric PDEs

Demonstrates neural network approximation capabilities

Validates effectiveness with numerical experiments

Abstract

Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It utilizes a separated representation form for the parametric PDE solution via a deep neural network and a least-squares solver. In this approach, the output of the deep neural network consists of a vector-valued function, interpreted as basis functions for the parametric solution space, and the least-squares solver determines the optimal solution within the constructed solution space for each given parameter. The LS-Net method requires a quadratic loss function for the least-squares solver to find optimal solutions given the set of basis functions. In this study, we consider loss functions derived from the Deep Fourier Residual and Physics-Informed Neural Networks approaches. We also provide theoretical results similar to the Universal Approximation Theorem, stating that there exists a sufficiently large neural network that can theoretically approximate solutions of parametric PDEs with the desired accuracy. We illustrate the LS-net method by solving one- and two-dimensional problems. Numerical results clearly demonstrate the method's ability to approximate parametric solutions.