Evidential Physics-Informed Neural Networks for Scientific Discovery

TL;DR

E-PINN delivers an uncertainty-aware physics-informed neural network by fusing Deep Evidential Regression with PINN residuals to infer unknown PDE parameters via a learned posterior. The framework introduces an adaptive PDE residual weight and a data-driven prior for parameters, implemented in two training phases. Validation on Poisson and Fisher-KPP PDEs demonstrates superior calibration of uncertainty (empirical coverage probability) compared to Bayesian PINN and Deep Ensemble, while a targeted Bergman minimal model application shows clinically relevant parameter indices can be inferred with uncertainty that aids discrimination between health states. The approach is broadly applicable to inverse problems in complex PDE systems and offers a principled path for quantified discovery in scientific domains.

Abstract

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.

Figures (4)

• Figure 1: The left figure shows the model prediction enclosed within a 95% confidence band, together with the exact solution and training dataset. The right ECP vs NCP plot shows the model's uncertainty distribution to be well-calibrated.
• Figure 2: Heatmap representation of the model's prediction displayed next to the training dataset for comparison. Noise was added to the region bounded within the red dashed lines. The rightmost diagram shows the uncertainty distribution heatmap mirroring the noisy region in the training data.
• Figure 3: Each row shows the model prediction and its associated uncertainty distributions for the B-PINN (top) and Deep Ensemble (bottom) methods. There is an absence of elevated uncertainty values within the red boundaries (where noise was inserted in the training data) unlike the case of E-PINN (Fig. 2), showing low correlation between uncertainty estimates and the added noise for these two methods.
• Figure 4: The left diagrams show E-PINN's prediction curves for glucose evolution in healthy and diabetic subjects. Blue dashed curves are the numerical solutions to Bergman's ODE with parameters $p_1, p_2, p_3$ being the posterior modes. Deviations between blue and red curves in the middle regions suggest that PDE description may be relatively less valid during those durations. The right diagram superposes the learned posterior distributions of the indices $S_G, S_I$, revealing the latter as a more discriminatory index for identifying diabetic individuals. Log scale for $S_I$ was adopted to enable clearer visualization of the distributions for healthy (dark brown) and diabetic (light brown) patients.