B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data

过度拟合 不确定度量化 贝叶斯概率 反问题 后验概率 估计员 人工神经网络 贝叶斯推理 贝叶斯线性回归 计算机科学 偏微分方程 算法 人工智能 数学 数学优化 应用数学 机器学习 统计 数学分析
作者
Liu Yang,Xuhui Meng,George Em Karniadakis
出处
期刊:Journal of Computational Physics [Elsevier BV]
卷期号:425: 109913-109913 被引量:567
标识
DOI:10.1016/j.jcp.2020.109913
摘要

We propose a Bayesian physics-informed neural network (B-PINN) to solve both forward and inverse nonlinear problems described by partial differential equations (PDEs) and noisy data. In this Bayesian framework, the Bayesian neural network (BNN) combined with a PINN for PDEs serves as the prior while the Hamiltonian Monte Carlo (HMC) or the variational inference (VI) could serve as an estimator of the posterior. B-PINNs make use of both physical laws and scattered noisy measurements to provide predictions and quantify the aleatoric uncertainty arising from the noisy data in the Bayesian framework. Compared with PINNs, in addition to uncertainty quantification, B-PINNs obtain more accurate predictions in scenarios with large noise due to their capability of avoiding overfitting. We conduct a systematic comparison between the two different approaches for the B-PINNs posterior estimation (i.e., HMC or VI), along with dropout used for quantifying uncertainty in deep neural networks. Our experiments show that HMC is more suitable than VI with mean field Gaussian approximation for the B-PINNs posterior estimation, while dropout employed in PINNs can hardly provide accurate predictions with reasonable uncertainty. Finally, we replace the BNN in the prior with a truncated Karhunen-Loève (KL) expansion combined with HMC or a deep normalizing flow (DNF) model as posterior estimators. The KL is as accurate as BNN and much faster but this framework cannot be easily extended to high-dimensional problems unlike the BNN based framework.
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