The curse of dimensionality: what lies beyond the capabilities of physics-informed neural networks

TL;DR

This paper investigates the limitations of physics-informed neural networks (PINNs), revealing that they struggle with inverse problems involving multiple parameters, thus defining their applicability boundaries in physical system analysis.

Contribution

The study demonstrates the fundamental limitations of PINNs in inverse problems with multiple parameters using a simple RC filter example, highlighting their boundaries.

Findings

PINNs accurately predict forward system dynamics.

PINNs fail to recover unique parameters in inverse problems with more than two parameters.

The results clarify the applicability limits of PINNs for parameter discovery.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a promising framework for solving forward and inverse problems governed by differential equations. However, their reliability when used in ill-posed inverse problems remains poorly understood. In this study, we explore the fundamental limitations of PINNs using a simple illustrative case: RC low-pass filters. Showing that while PINNs can accurately predict system dynamics in forward problems, they fail to recover unique physical parameters when solving inverse problems when more than two parameters are approximated. Our findings provide grounds to understand the boundaries of PINNs applicability for parameter discovery in physical systems.