We consider the simulation of Bayesian statistical inverse problems governed by large-scale linear and nonlinear partial differential equations (PDEs). Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve such problems. However, MCMC techniques are computationally challenging as they require several thousands of forward PDE solves. The goal of this paper is to introduce a fractional deep neural network based approach for the forward solves within an MCMC routine. Moreover, we discuss some approximation error estimates and illustrate the efficiency of our approach via several numerical examples.

翻译：我们认为模拟巴伊西亚统计逆向问题是由大规模线性和非线性部分差异方程式(PDEs)所决定的。Markov链Monte Carlo(MCMC)算法是解决这些问题的标准技术。然而,MCMC技术在计算上具有挑战性,因为它们需要数千个前方PDE解决方案。本文的目的是在MCMC常规中引入一个基于分数深度神经网络的远方解决方案方法。此外,我们讨论一些近似误差估计,并通过几个数字例子说明我们方法的效率。