Evidential Physics-Informed Neural Networks for Scientific Discovery
This work addresses uncertainty estimation in scientific discovery for domains like physics and medicine, but it is incremental as it builds on existing PINN and evidential deep learning frameworks.
The paper tackled the problem of uncertainty quantification in physics-informed neural networks (PINNs) by introducing Evidential Physics-Informed Neural Networks (E-PINNs), which improved calibration of empirical coverage probabilities compared to Bayesian PINN and Deep Ensemble methods in case studies like the 1D Poisson and 2D Fisher-KPP equations.
We present the fundamental theory and implementation guidelines underlying Evidential Physics-Informed Neural Network (E-PINN) -- a novel class of uncertainty-aware PINN. It leverages the marginal distribution loss function of evidential deep learning for estimating uncertainty of outputs, and infers unknown parameters of the PDE via a learned posterior distribution. Validating our model on two illustrative case studies -- the 1D Poisson equation with a Gaussian source and the 2D Fisher-KPP equation, we found that E-PINN generated empirical coverage probabilities that were calibrated significantly better than Bayesian PINN and Deep Ensemble methods. To demonstrate real-world applicability, we also present a brief case study on applying E-PINN to analyze clinical glucose-insulin datasets that have featured in medical research on diabetes pathophysiology.