Variational Inference and Probabilistic Models for Parametric Partial Differential Equations

Parametric partial differential equations (PDEs) are of central importance to modern engineering sciences. They are the means for understanding the physical behaviour of systems for ranges of configurations and designs. The tools developed over the last few decades, such as finite elements, and finite volume methods are highly effective for single solution scenarios. However, these methods are ill-equipped in dealing with parametric problems as there is no information carry-over from one simulation to the next. This thesis is an attempt at adapting methods of probabilistic machine learning to create methodological advances in solving various problems relating to PDEs though variational inference and probabilistic models. The work is composed of three contributions. The first contribution lies in creating active learning surrogates for Bayesian inverse problems called Active learning projected surrogates – SVGD. Here we leverage Stein variational gradient descent methods to move clusters of particles through the information given by an active learning Gaussian process surrogate of the posterior surface posed by the classic Bayesian inverse problem framework. The second contribution comes from the development of a physics driven deep latent variable model. The developed variational inference framework leverages a virtual observable of a physics residual to inform the learning of jointly trained forward and inverse parametric PDE emulators. We implement our method with both finite element and physics informed neural network based residuals. The third contribution is the fundamental extension of the physics driven deep latent variable model into a function space view, inspired by neural processes we propose a variational inference framework leveraging random grids and Gaussian processes for parametric PDEs. The method treats partitions of PDE domains as random variables to be marginalized in training. Through this, we arrive at a random collocation method, shown to out-perform fixed grid methods. To implement this framework we also propose a new neural network architecture capable of effectively dealing with information given on random grids. We test our proposed methodologies on various PDEs relevant to engineering such as linear and nonlinear Poisson, nonlinear heat diffusion, thin-shells, wave, Burgers, and incompressible Navier-Stokes equations.La Fondation Baxter & Alma Ricar