  We present two approximate Bayesian inference methods for parameter
estimation in partial differential equation (PDE) models with space-dependent
and state-dependent parameters. We demonstrate that these methods provide
accurate and cost-effective alternatives to Markov Chain Monte Carlo
simulation. We assume a parameterized Gaussian prior on the unknown functions,
and approximate the posterior density by a parameterized multivariate Gaussian
density. The parameters of the prior and posterior are estimated from sparse
observations of the PDE model's states and the unknown functions themselves by
maximizing the evidence lower bound (ELBO), a lower bound on the log marginal
likelihood of the observations. The first method, Laplace-EM, employs the
expectation maximization algorithm to maximize the ELBO, with a Laplace
approximation of the posterior on the E-step, and minimization of a
Kullback-Leibler divergence on the M-step. The second method, DSVI-EB, employs
the doubly stochastic variational inference (DSVI) algorithm, in which the ELBO
is maximized via gradient-based stochastic optimization, with nosiy gradients
computed via simple Monte Carlo sampling and Gaussian backpropagation. We apply
these methods to identifying diffusion coefficients in linear and nonlinear
diffusion equations, and we find that both methods provide accurate estimates
of posterior densities and the hyperparameters of Gaussian priors. While the
Laplace-EM method is more accurate, it requires computing Hessians of the
physics model. The DSVI-EB method is found to be less accurate but only
requires gradients of the physics model.
