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Physics-informed neural network (PINN) has been successfully applied insolving a variety of nonlinear non-convex forward and inverse problems.However, the training is challenging because of the non-convex loss functionsand the multiple optima in the Bayesian inverse problem. In this work, wepropose a multi-variance replica exchange stochastic gradient Langevindiffusion method to tackle the challenge of the multiple local optima in theoptimization and the challenge of the multiple modal posterior distribution inthe inverse problem. Replica exchange methods are capable of escaping from thelocal traps and accelerating the convergence. However, it may not be efficientto solve mathematical inversion problems by using the vanilla replica methoddirectly since the method doubles the computational cost in evaluating theforward solvers (likelihood functions) in the two chains. To address thisissue, we propose to make different assumptions on the energy functionestimation and this facilities one to use solvers of different fidelities inthe likelihood function evaluation. We give an unbiased estimate of theswapping rate and give an estimation of the discretization error of the scheme.To verify our idea, we design and solve four inverse problems which havemultiple modes. The proposed method is also employed to train the Bayesian PINNto solve the forward and inverse problems; faster and more accurate convergencehas been observed when compared to the stochastic gradient Langevin diffusion(SGLD) method and vanila replica exchange methods.

Multi-variance replica exchange stochastic gradient MCMC for inverse and forward Bayesian physics-informed neural network