Optimizing Variational Physics-Informed Neural Networks Using Least Squares

TL;DR

This paper introduces a hybrid Least Squares and gradient descent method to improve convergence in Variational Physics-Informed Neural Networks, offering efficient implementation strategies and demonstrating significant speedups in numerical experiments.

Contribution

It proposes a novel hybrid optimizer combining Least Squares with gradient descent for PINNs, along with efficient implementation techniques to reduce computational costs.

Findings

Hybrid Least Squares/Gradient Descent optimizer improves convergence.

Forward-mode automatic differentiation speeds up training by up to 100 times.

Numerical experiments validate the efficiency and effectiveness of the proposed methods.

Abstract

Variational Physics-Informed Neural Networks often suffer from poor convergence when using stochastic gradient-descent-based optimizers. By introducing a Least Squares solver for the weights of the last layer of the neural network, we improve the convergence of the loss during training in most practical scenarios. This work analyzes the computational cost of the resulting hybrid Least-Squares/Gradient-Descent optimizer and explains how to implement it efficiently. In particular, we show that a traditional implementation based on backward-mode automatic differentiation leads to a prohibitively expensive algorithm. To remedy this, we propose using either forward-mode automatic differentiation or an ultraweak-type scheme that avoids the differentiation of trial functions in the discrete weak formulation. The proposed alternatives are up to one hundred times faster than the traditional one, recovering a computational cost-per-iteration similar to that of a conventional gradient-descent-based optimizer alone. To support our analysis, we derive computational estimates and conduct numerical experiments in one- and two-dimensional problems.