Bayesian Reasoning for Physics Informed Neural Networks
Krzysztof M. Graczyk Kornel Witkowski
Physics informed neural network (PINN) approach in Bayesian formulation is
presented. We adopt the Bayesian neural network framework formulated by MacKay
(Neural Computation 4 (3) (1992) 448). The posterior densities are obtained
from Laplace approximation. For each model (fit), the so-called evidence is
computed. It is a measure that classifies the hypothesis. The most optimal
solution has the maximal value of the evidence. The Bayesian framework allows
us to control the impact of the boundary contribution to the total loss.
Indeed, the relative weights of loss components are fine-tuned by the Bayesian
algorithm. We solve heat, wave, and Burger's equations. The obtained results
are in good agreement with the exact solutions. All solutions are provided with
the uncertainties computed within the Bayesian framework.