Uncertainty quantification of data-driven stochastic PDE with physics-informed variational inference Pohang University of Science and Technology

Abstract

Stochastic partial differential equations (SPDEs) are partial differential equations (PDEs) with random parameters. Even though efficient numerical methods have been developed to solve SPDE for decades, it is still challenging to deal with SPDE when high-dimensional random parameters are involved. This doctoral dissertation proposes to use deep learning based approach, called a physics-informed variational inference (PIVI) for uncertainty quantification of data-driven SPDE problems. We aim to provide a start point to deal with high-dimensional SPDE problems with deep learning based approach by integrating physics-informed machine learning method into variational inference framework. In PIVI, the target function is described by two neural networks; one is a deterministic function of spatial variables, and another represents its random coefficients. Hence, the target function is expressed by the inner product of two networks. We employ another neural network to approximate the posterior distribution of the random coefficients given an observation. Then the three networks are trained by maximizing evidence lower bound (ELBO), which incorporates the given physical laws. We also show that the derived ELBO consists of physics-informed loss and regularization terms for the random coefficients. After the training, PIVI efficiently generates a realization of target function just by forward passes of neural networks. We numerically demonstrate the efficiency of PIVI with 3 examples: learning stochastic process, solving Poisson equation, and solving elliptic equations. In the first example, we study about the effect of hyper parameters of PIVI, demonstrating that PIVI efficiently learns and successfully reconstructs high-dimensional random processes. In the second example, we use PIVI to solve a forward problem of a Poisson equation where source term is random, explaining that physics-informed learning scheme can be integrated into variational inference framework, providing possibility of PIVI to addresses SPDE problems. In the third example, we show how PIVI can be used to forward and inverse problem of an elliptic equation, validating that PIVI can be applied to forward and inverse problem in a unified framework, even though the equation is high-dimensional.