We develop a new Bayesian framework based on deep neural networks to be able to
extrapolate in space-time using historical data and to quantify uncertainties arising from
both noisy and gappy data in physical problems. Specifically, the proposed approach has
two stages: (1) prior learning and (2) posterior estimation. At the first stage, we employ the
physics-informed Generative Adversarial Networks (PI-GAN) to learn a functional prior either
from a prescribed function distribution, e.g., Gaussian process, or from historical data and
physics. At the second stage, we employ the Hamiltonian Monte Carlo (HMC) method to
estimate the posterior in the latent space of PI-GANs. In addition, we use two different
approaches to encode the physics: (1) automatic differentiation, used in the physicsinformed neural networks (PINNs) for scenarios with explicitly known partial differential
equations (PDEs), and (2) operator regression using the deep operator network (DeepONet)
for PDE-agnostic scenarios. We then test the proposed method for (1) meta-learning for
one-dimensional regression, and forward/inverse PDE problems (combined with PINNs);
(2) PDE-agnostic physical problems (combined with DeepONet), e.g., fractional diffusion
as well as saturated stochastic (100-dimensional) flows in heterogeneous porous media;
and (3) spatial-temporal regression problems, i.e., inference of a marine riser displacement
field using experimental data from the Norwegian Deepwater Programme (NDP). The
results demonstrate that the proposed approach can provide accurate predictions as well
as uncertainty quantification given very limited scattered and noisy data, since historical
data could be available to provide informative priors. In summary, the proposed method is
capable of learning flexible functional priors, e.g., both Gaussian and non-Gaussian process,
and can be readily extended to big data problems by enabling mini-batch training using
stochastic HMC or normalizing flows since the latent space is generally characterized as low dimensional.
Published: November 9, 2022
Citation
Meng X., L. Yang, Z. Mao, J. Del Aguila Ferrandis, G.E. Karniadakis, and G.E. Karniadakis. 2022.Learning Functional Priors and Posteriors from Data and Physics.Journal of Computational Physics 457.PNNL-SA-178723.doi:10.1016/j.jcp.2022.111073