Uncertainty Quantification for Physics-Informed Neural Networks with Extended Fiducial Inference

Frank Shih, Zhenghao Jiang, Faming Liang

Keywords: Adaptive Stochastic Gradient MCMC, Extended Fiducial Inference, Physics-Informed Neural Network, Stochastic Neural Network, Uncertainty Quantification

Abstract:

Uncertainty quantification (UQ) in scientific machine learning is increasingly
critical as neural networks are widely adopted to tackle complex
problems across diverse scientific disciplines.
For physics-informed neural networks (PINNs), a prominent model in scientific machine learning, uncertainty is typically quantified using
Bayesian or dropout methods. However, both approaches suffer from a fundamental limitation: the prior distribution or dropout rate required to construct
honest confidence sets cannot be determined without additional information.
In this paper, we propose a novel method within the framework of extended fiducial inference (EFI) to provide rigorous uncertainty quantification for PINNs. The proposed method leverages a narrow-neck hyper-network to learn the parameters of the PINN and quantify their uncertainty based on imputed random errors in the observations. This approach overcomes the limitations of Bayesian and dropout methods, enabling the construction of honest confidence sets based solely on observed data.
This advancement represents a significant breakthrough for PINNs, greatly enhancing their reliability, interpretability, and applicability to real-world scientific and engineering challenges. Moreover, it establishes
a new theoretical framework for EFI, extending its application to large-scale models, eliminating the need for sparse hyper-networks, and significantly improving the automaticity and robustness of statistical inference.

Primary Area: Deep learning (e.g., architectures, generative models, optimization for deep networks, foundation models, LLMs)

Submission Number: 13444