Past Event: Oden Institute Seminar

Efficient geometric Markov chain Monte Carlo for Bayesian inversion enabled by derivative-informed neural operator

Lianghao Cao, Oden Institute for Computational Engineering and Sciences

Abstract

Markov chain Monte Carlo (MCMC) allows one to asymptotically exactly sample from an unnormalized target probability distribution, making it the gold standard for rigorous solution of Bayesian inverse problems. For a high-dimensional target probability distribution with complex geometry, the choice of the proposal distribution in a MCMC method greatly affects its sampling efficiency. A moving proposal that adapts to the local geometry of the target makes MCMC highly efficient. However, sampling from such a proposal requires derivative and Hessian information about the target, which can be computationally expensive when the target is defined through large-scale partial differential equations (PDEs).

This talk presents an operator learning approach to accelerate MCMC for infinite-dimensional nonlinear Bayesian inversion. First, we construct a surrogate of the nonlinear parameter-to-observation map via training a derivative-informed neural operator (DINO). The learned operator surrogate is then used to generate position-dependent proposals for function space geometric MCMC with delayed acceptance. DINO is trained via approximation error control in the topology of H^1 Sobolev space with Gaussian measure. Compared to neural operators trained with L^2 approximation error control, DINO possesses a much more accurate derivative and requires a smaller training sample size to achieve the same L^2 accuracy. As a result, DINO enables (i) rapid and accurate approximate sampling from local Gaussian approximations of the posterior distribution, and (ii) a highly efficient delayed acceptance procedure. Preliminary numerical results for a nonlinear diffusion-reaction PDE problem show dramatic computational speedups for MCMC sampling using our approach over existing MCMC methods with fixed and moving proposals.

The presented results are joint work with Thomas O’Leary-Roseberry and Omar Ghattas.

Biography

Lianghao Cao is a Postdoctoral Fellow at the Oden Institute for Computational Engineering and Sciences, the University of Texas at Austin. He completed his PhD under the CSEM program at the Oden Institute in 2022. He obtained a B.S. in Engineering Mechanics from University of Illinois at Urbana-Champaign in 2017. His research interest is Bayesian uncertainty quantification and optimization for systems governed by partial differential equations.