B-PINNs: Bayesian Physics-informal Neural Networks for Forward and Inverse PDE Problems with Noisy Data
- Brown University
- BROWN UNIVERSITY
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.
- Research Organization:
- Pacific Northwest National Laboratory (PNNL), Richland, WA (United States)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- AC05-76RL01830
- OSTI ID:
- 1763962
- Report Number(s):
- PNNL-SA-158354
- Journal Information:
- Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 425
- Country of Publication:
- United States
- Language:
- English
| Physics-aware deep neural networks for surrogate modeling of turbulent natural convection | preprint | January 2021 |
| An extended physics informed neural network for preliminary analysis of parametric optimal control problems | preprint | January 2021 |
| Novel Deep neural networks for solving Bayesian statistical inverse | preprint | January 2021 |
| A Machine-Learning Method for Time-Dependent Wave Equations over Unbounded Domains | preprint | January 2021 |
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