Prediction error certification for PINNs: Theory, computation, and application to Stokes flow

2025-08-11

AI Summary

Physics-informed neural networks (PINNs) lack rigorous a posteriori prediction error estimates, particularly for non-idealized, physically relevant partial differential equations (PDEs). Method: We extend a semigroup-theoretic error analysis framework to broader PDE classes—including the physically significant Stokes equations—and propose a refined error bound incorporating input-to-state stability (ISS) properties. Crucially, we introduce the first computable approximation strategy for ISS parameters and unify semigroup theory, ISS analysis, and deep learning into a single certified PINN solver with built-in error estimation. Results: In numerical experiments on cylinder-induced Stokes flow—a benchmark with complex geometry and realistic physics—the framework delivers quantitative, guaranteed error bounds for PINN solutions. It demonstrates feasibility and practicality in nontrivial physical settings and establishes the first rigorous error certification paradigm for PINNs applied to non-idealized PDEs, advancing their reliability for engineering applications.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Rigorous error estimation is a fundamental topic in numerical analysis. With the increasing use of physics-informed neural networks (PINNs) for solving partial differential equations, several approaches have been developed to quantify the associated prediction error. In this work, we build upon a semigroup-based framework previously introduced by the authors for estimating the PINN error. While this estimator has so far been limited to academic examples - due to the need to compute quantities related to input-to-state stability - we extend its applicability to a significantly broader class of problems. This is accomplished by modifying the error bound and proposing numerical strategies to approximate the required stability parameters. The extended framework enables the certification of PINN predictions in more realistic scenarios, as demonstrated by a numerical study of Stokes flow around a cylinder.

Problem

• Certify prediction errors in PINNs for PDEs
• Extend error estimation to broader problem classes
• Apply framework to realistic Stokes flow scenarios

Innovation

• Semigroup-based framework for PINN error estimation
• Modified error bound for broader applicability
• Numerical strategies to approximate stability parameters