Regularity and error estimates in physics-informed neural networks for the Kuramoto-Sivashinsky equation

TL;DR

This paper develops rigorous error estimates for physics-informed neural networks approximating the complex, nonlinear Kuramoto-Sivashinsky equation, combining mathematical analysis with numerical validation to address longstanding challenges.

Contribution

It establishes the first rigorous error bounds for PINNs applied to the Kuramoto-Sivashinsky equation, advancing theoretical understanding and practical application.

Findings

Derived global regularity criteria in Besov spaces.

Established the first rigorous error estimates for PINNs on this equation.

Validated theoretical bounds through numerical experiments.

Abstract

Due to its nonlinearity, bi-harmonic dissipation, and backward heat-like term in the absence of a divergence-free condition, the $2$-D/$3$-D Kuramoto-Sivashinsky equation poses significant challenges for both mathematical analysis and numerical approximation. These difficulties motivate the development of methods that blend classical analysis with numerical approximation approaches embodied in the framework of the physics-informed neural networks (PINNs). In addition, despite the extensive use of PINN frameworks for various linear and nonlinear PDEs, no study had previously established rigorous error estimates for the Kuramoto-Sivashinsky equation within a PINN setting. In this work, we overcome the inherent challenges, and establish several global regularity criteria based on space-time integrability conditions in Besov spaces. We then derive the first rigorous error estimates for the PINNs approximation of the Kuramoto-Sivashinsky equation and validate our theoretical error bounds through numerical simulations.