Uncertainty Quantification for Physics-Informed Neural Networks with Extended Fiducial Inference
Frank Shih, Zhenghao Jiang, Faming Liang
TL;DR
This work addresses the challenge of honest uncertainty quantification for physics-informed neural networks (PINNs) by introducing extended fiducial inference (EFI). EFI reframes observations as data-generating equations with latent errors and learns an inverse mapping ${oldsymbol heta}=G({oldsymbol Y}_n,{oldsymbol X}_n,{oldsymbol Z}_n)$ using a two-network architecture, enabling fiducial sampling of model parameters without requiring priors. A pivotal contribution is the new theoretical framework employing a narrow-neck ${oldsymbol w}$-network (and StoNet concepts) that permits consistent inference in large-scale models where ${oldsymbol heta}$ is high-dimensional, thereby extending EFI to PINNs and large neural nets. Empirical results on synthetic 1-D Poisson problems and real data (Montroll growth and FKPP-type models) demonstrate that EFI achieves honest confidence sets with near-nominal coverage and tight intervals, outperforming Bayesian PINNs and standard dropout-based approaches in terms of calibration and robustness. The work provides a principled, data-driven pathway to reliable UQ in scientific ML, with implications for interpretability and deployment in real-world modeling tasks, and outlines future directions for scalability and transfer learning.
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.