Transforming physics-informed machine learning to convex optimization

TL;DR

This paper introduces Convex-PIML, a framework that transforms physics-informed machine learning into convex optimization problems, enabling more efficient and reliable solutions for scientific modeling.

Contribution

It develops a novel convexification approach for PIML using B-splines and adaptive knot optimization, overcoming optimization challenges in traditional PIML methods.

Findings

Effective solution of diverse physical problems

Improved performance with adaptive knot optimization

Theoretically guaranteed convex optimization framework

Abstract

Physics-Informed Machine Learning (PIML) offers a powerful paradigm of integrating data with physical laws to address important scientific problems, such as parameter estimation, inferring hidden physics, equation discovery, and state prediction, etc. However, PIML still faces many serious optimization challenges that significantly restrict its applications. In this study, we propose a comprehensive framework that transforms PIML to convex optimization to overcome all these limitations, referred to as Convex-PIML. The linear combination of B-splines is utilized to approximate the data, promoting the convexity of the loss function. By replacing the non-convex components of the loss function with convex approximations, the problem is further converted into a sequence of successively refined approximated convex optimization problems. This conversion allows the use of well-established convex optimization algorithms, obtaining solutions effectively and efficiently. Furthermore, an adaptive knot optimization method based on error estimate is introduced to mitigate the spectral bias issue of PIML, further improving the performance. The proposed theoretically guaranteed framework is tested in scenarios with distinct types of physical prior. The results indicate that optimization problems are effectively solved in these scenarios, highlighting the potential of the framework for broad applications.