Building Trust in PINNs: Error Estimation through Finite Difference Methods

2026-03-16

AI Summary

This work addresses the challenge that physics-informed neural networks (PINNs) often lack reliable pointwise error estimates, thereby limiting the credibility of their predictions. To overcome this limitation, the authors propose a lightweight post-processing approach that models the PINN error as a partial differential equation governed by the same differential operator as the original problem. This error equation is driven by the residual of the PINN solution and solved using finite difference methods, enabling the generation of highly accurate and interpretable pointwise error maps without requiring access to the true solution. Evaluated across multiple standard PDE benchmarks, the method significantly enhances the verifiability and reliability of PINN predictions with minimal computational overhead.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their flexibility, PINNs offer limited insight into how their predictions deviate from the true solution, hindering trust in their prediction quality. We propose a lightweight post-hoc method that addresses this gap by producing pointwise error estimates for PINN predictions, which offer a natural form of explanation for such models, identifying not just whether a prediction is wrong, but where and by how much. For linear partial differential equations, the error between a PINN approximation and the true solution satisfies the same differential operator as the original problem, but driven by the PINN's PDE residual as its source term. We solve this error equation numerically using finite difference methods requiring no knowledge of the true solution. Evaluated on several benchmark PDEs, our method yields accurate error maps at low computational cost, enabling targeted and interpretable validation of PINNs.

Problem

• Physics-informed neural networks
• error estimation
• partial differential equations
• trust
• prediction reliability

Innovation

• Physics-informed neural networks
• Error estimation
• Finite difference methods
• PDE residual
• Interpretable validation