MCMC-PINNs: Adaptively Sampling Collocation Points of PINNs with an Improved Markov Chain Monte Carlo Method

Source abstract

In recent years, physics-informed neural networks (PINNs) have emerged as an effective method for solving partial differential equations (PDEs). PINNs offer several advantages such as no limitation of dimensionality, low data requirements, and simple inverse problem solving. Introducing collocation points and PDE loss is the key to PINNs, while the distribution of collocation points significantly impacts the training efficiency of PINNs. Uniformly distributed collocation points may not capture the different-scale features of the solution. To address this problem, we propose an adaptive method for sampling the collocation points using an improved Markov Chain Monte Carlo method (MCMC). The MCMC-based adaptive collocation point sampling method (MCMC-PINNs) contain two key components: constructing the distribution of collocation points and sampling it effectively. The residual has been validated as an indicator for collocation point distribution, and the residual of PDE converges to $0$ as training progresses. Thus, we choose a monotonically increasing function of the residual as the unnormalized probability distribution of the collocation points and use MCMC to sample the collocation points. In this paper, we present the convergence analysis of MCMC-PINNs and prove the error bound of it. Finally, several numerical examples are used to illustrate the performance of the MCMC-PINNs.