Quantifying uncertainty in physics-informed neural networks

Abstract

The integration of machine learning in the development of digital twins has seen significant advancements, with neural networks playing a key role due to their ability to approximate complex patterns. However, their "black-box" nature poses challenges in interpretability and validation. Physics-informed neural networks (PINNs) address this by incorporating known laws of the system into the training of the neural networks. Despite its promise, current PINN implementations lack standardization, making them difficult to reproduce and evaluate comprehensively.

This paper introduces a standardized abstraction for PINNs within the Kedro framework to streamline implementation and improve reproducibility.

We also explore uncertainty quantification (UQ) methods to assess model robustness beyond traditional error-based metrics. In this domain, we introduce PINN-DER, a method that adapts evidential learning to learn uncertainty estimates alongside prediction. We observed on the Burgers and Laplace equations that PINN-DER reduced maximum error while maintaining predictive accuracy. Our approach provides a structured foundation for integrating PINNs with UQ, improving its reliability in data-driven applications.