# Approximate Bayesian Model Inversion for PDEs with Heterogeneous and State-Dependent Coefficients

## TL;DR

This paper introduces two approximate Bayesian inference methods for PDE parameter estimation with space- and state-dependent coefficients, offering accurate and computationally efficient alternatives to traditional MCMC techniques.

## Contribution

The paper develops two novel variational Bayesian methods, Laplace-EM and DSVI-EB, for efficient parameter inference in PDE models with complex coefficients, improving computational cost and accuracy.

## Key findings

- Both methods accurately estimate posterior densities and hyperparameters.
- Laplace-EM provides higher accuracy but requires Hessian computations.
- DSVI-EB is less accurate but only needs gradient information.

## Abstract

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.