Evidential Physics-Informed Neural Networks

We present a novel class of Physics-Informed Neural Networks that is formulated based on the principles of Evidential Deep Learning, where the model incorporates uncertainty quantification by learning parameters of a higher-order distribution. The dependent and trainable variables of the PDE residual loss and data-fitting loss terms are recast as functions of the hyperparameters of an evidential prior distribution. Our model is equipped with an information-theoretic regularizer that contains the Kullback-Leibler divergence between two inverse-gamma distributions characterizing predictive uncertainty. Relative to Bayesian-Physics-Informed-Neural-Networks, our framework appeared to exhibit higher sensitivity to data noise, preserve boundary conditions more faithfully and yield empirical coverage probabilities closer to nominal ones. Toward examining its relevance for data mining in scientific discoveries, we demonstrate how to apply our model to inverse problems involving 1D and 2D nonlinear differential equations.