Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Mohammad Mahabubur Rahman, Deepanshu Verma

TL;DR

The paper addresses the challenging Kuramoto-Sivashinsky equation in 2D/3D by deriving global regularity criteria within Besov spaces and establishing the first rigorous $L^2$ error estimates for physics-informed neural networks approximating this equation. It integrates Prodi-Serrin-type space-time integrability conditions with a PINN residual framework to bound the training and generalization errors, and shows how these bounds translate into a computable $L^2$ error between the PINN solution and the true solution. The authors demonstrate that, under appropriate Besov-space conditions on the initial data and solution, PINN residuals control the $L^2$-error, and provide explicit decay rates in terms of quadrature sizes and network capacity. Numerical experiments in two dimensions corroborate the theory, revealing a plateau behavior due to logarithmic factors and highlighting the interplay between quadrature resolution and network approximation in achieving accurate PINN solutions. The results advance the theoretical understanding of PINNs for chaotic, higher-order PDEs and offer practical guidance for error control in physics-informed learning in periodic domains.

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.

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

TL;DR

The paper addresses the challenging Kuramoto-Sivashinsky equation in 2D/3D by deriving global regularity criteria within Besov spaces and establishing the first rigorous error estimates for physics-informed neural networks approximating this equation. It integrates Prodi-Serrin-type space-time integrability conditions with a PINN residual framework to bound the training and generalization errors, and shows how these bounds translate into a computable error between the PINN solution and the true solution. The authors demonstrate that, under appropriate Besov-space conditions on the initial data and solution, PINN residuals control the -error, and provide explicit decay rates in terms of quadrature sizes and network capacity. Numerical experiments in two dimensions corroborate the theory, revealing a plateau behavior due to logarithmic factors and highlighting the interplay between quadrature resolution and network approximation in achieving accurate PINN solutions. The results advance the theoretical understanding of PINNs for chaotic, higher-order PDEs and offer practical guidance for error control in physics-informed learning in periodic domains.

Abstract

Due to its nonlinearity, bi-harmonic dissipation, and backward heat-like term in the absence of a divergence-free condition, the -D/-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.

Paper Structure

This paper contains 17 sections, 23 theorems, 103 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Kuramoto-Sivashinsky Equation
  3. Physics-Informed Neural Networks
  4. Plan for the Paper:
  5. Preliminaries
  6. Notations and Spaces
  7. Besov Spaces
  8. The Concepts of Solutions
  9. Neural Networks
  10. Quadrature rules
  11. Physics-informed neural networks
  12. Main Results
  13. The Regularity Results
  14. Error Analysis
  15. Numerical Experiments
  16. ...and 2 more sections

Key Result

Lemma 2.2

Let $u$ be a strong solution, then $\partial_t u, (u \cdot \nabla)u, \Delta u, \textrm{and}\; \Delta^2 u \in L^2([0, T); L^2(\mathbb{T}^d))$.

Figures (4)

  • Figure 4.1: PINNs Solution, Manufacture (Exact) Solution, and Absolute Error
  • Figure 4.2: PINNs Solution, Manufacture (Exact) Solution, and Absolute Error
  • Figure 4.3: Training error, total error, and bound for different numbers of residual points
  • Figure 4.4: Training error, total error, and bound for different numbers of neurons

Theorems & Definitions (44)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • ...and 34 more