Bayesian Reasoning for Physics Informed Neural Networks
This work addresses uncertainty estimation in PINNs for physics-based modeling, which is incremental as it adapts existing Bayesian methods to a specific framework.
The authors tackled the problem of incorporating uncertainty quantification into physics-informed neural networks (PINNs) by applying a Bayesian formulation, resulting in predictions comparable to Hybrid Monte Carlo sampling and accurate solutions for heat, wave, and Burger's equations with computed uncertainties.
We present the application of the physics-informed neural network (PINN) approach in Bayesian formulation. We have adopted the Bayesian neural network framework to obtain posterior densities from Laplace approximation. For each model or fit, the evidence is computed, which is a measure that classifies the hypothesis. The optimal solution is the one with the highest value of evidence. We have proposed a modification of the Bayesian algorithm to obtain hyperparameters of the model. We have shown that within the Bayesian framework, one can obtain the relative weights between the boundary and equation contributions to the total loss. Presented method leads to predictions comparable to those obtained by sampling from the posterior distribution within the Hybrid Monte Carlo algorithm (HMC). We have solved heat, wave, and Burger's equations, and the results obtained are in agreement with the exact solutions, demonstrating the effectiveness of our approach. In Burger's equation problem, we have demonstrated that the framework can combine information from differential equations and potential measurements. All solutions are provided with uncertainties (induced by the model's parameter dependence) computed within the Bayesian framework.