Certified machine learning: Rigorous a posteriori error bounds for PDE defined PINNs

TL;DR

This paper introduces a rigorous a posteriori error bound for physics-informed neural networks (PINNs), enabling prediction error quantification without needing the true solution, based solely on known system characteristics.

Contribution

It provides the first rigorous upper bound on PINN prediction error that can be computed a posteriori without the true solution, applicable to various PDEs.

Findings

Error bounds successfully applied to transport, heat, Navier-Stokes, and Klein-Gordon equations.

The bounds are computable using only a priori information about the dynamical system.

Demonstrates the potential for rigorous error quantification in physics-informed machine learning.

Abstract

Prediction error quantification in machine learning has been left out of most methodological investigations of neural networks, for both purely data-driven and physics-informed approaches. Beyond statistical investigations and generic results on the approximation capabilities of neural networks, we present a rigorous upper bound on the prediction error of physics-informed neural networks. This bound can be calculated without the knowledge of the true solution and only with a priori available information about the characteristics of the underlying dynamical system governed by a partial differential equation. We apply this a posteriori error bound exemplarily to four problems: the transport equation, the heat equation, the Navier-Stokes equation and the Klein-Gordon equation.