We develop a novel deep learning method for uncertainty quantification in stochastic partial differential equations based on Bayesian neural network (BNN) and Hamiltonian Monte Carlo (HMC). A BNN efficiently learns the posterior distribution of the parameters in deep neural networks by performing Bayesian inference on the network parameters. The posterior distribution is efficiently sampled using HMC to quantify uncertainties in the system. Several numerical examples are shown for both forward and inverse problems in high dimension to demonstrate the effectiveness of the proposed method for uncertainty quantification. These also show promising results that the computational cost is almost independent of the dimension of the problem demonstrating the potential of the method for tackling the so-called curse of dimensionality.
翻译:我们根据Bayesian神经网络(BNN)和Hamiltonian Monte Carlo(HMC),开发了一种新型的深层学习方法,用于在随机部分差异方程式中量化不确定性。一个BNN通过对网络参数进行Bayesian推理,有效地了解深神经网络参数的后方分布。后方分布利用HMC有效抽样,以量化系统中的不确定性。一些高端前方和反面问题的数字实例显示了拟议不确定性量化方法的有效性。这些也显示了有希望的结果,即计算成本几乎独立于问题的层面,表明解决所谓维度诅咒的方法的潜力。