Unveiling the optimization process of Physics Informed Neural Networks: How accurate and competitive can PINNs be?

TL;DR

The paper presents elsarticle.cls, a modern LaTeX document class designed to format manuscripts for Elsevier journals with minimal package clashes. It explains the rationale for building on article.cls, contrasts it with the older elsart.cls, and outlines key enhancements in formatting flexibility, front matter handling, and integration with standard tooling such as natbib and hyperref. The authors provide installation guidance via CTAN and Elsevier resources and describe the recommended workflow for using preprint or final formats. The work aims to streamline manuscript preparation and ensure consistent, publication-ready formatting across Elsevier venues.

Abstract

This study investigates the potential accuracy boundaries of physics-informed neural networks, contrasting their approach with previous similar works and traditional numerical methods. We find that selecting improved optimization algorithms significantly enhances the accuracy of the results. Simple modifications to the loss function may also improve precision, offering an additional avenue for enhancement. Despite optimization algorithms having a greater impact on convergence than adjustments to the loss function, practical considerations often favor tweaking the latter due to ease of implementation. On a global scale, the integration of an enhanced optimizer and a marginally adjusted loss function enables a reduction in the loss function by several orders of magnitude across diverse physical problems. Consequently, our results obtained using compact networks (typically comprising 2 or 3 layers of 20-30 neurons) achieve accuracies comparable to finite difference schemes employing thousands of grid points. This study encourages the continued advancement of PINNs and associated optimization techniques for broader applications across various fields.