Efficient Bayesian multi-fidelity inverse analysis for expensive and non-differentiable physics-based simulations in high stochastic dimensions

TL;DR

This paper introduces Bayesian multi-fidelity inverse analysis (BMFIA), a method that efficiently performs high-dimensional Bayesian inverse problems using cheaper lower-fidelity models to approximate complex physics-based simulations.

Contribution

The paper presents a novel Bayesian inference approach that leverages multi-fidelity models to efficiently solve high-dimensional inverse problems without requiring derivatives of complex codes.

Findings

BMFIA effectively uses few high- and low-fidelity simulations (50-300) for learning.

The approach is fully differentiable and adaptable to various low-fidelity models.

It enables Bayesian inverse analysis in high stochastic dimensions where traditional methods are infeasible.

Abstract

High-dimensional Bayesian inverse analysis (dim >> 100) is mostly unfeasible for computationally demanding, nonlinear physics-based high-fidelity (HF) models. Usually, the use of more efficient gradient-based inference schemes is impeded if the multi-physics models are provided by complex legacy codes. Adjoint-based derivatives are either exceedingly cumbersome to derive or non-existent for practically relevant large-scale nonlinear and coupled multi-physics problems. Similarly, holistic automated differentiation w.r.t. primary variables of multi-physics codes is usually not yet an option and requires extensive code restructuring if not considered from the outset in the software design. This absence of differentiability further exacerbates the already present computational challenges. To overcome the existing limitations, we propose a novel inference approach called Bayesian multi-fidelity inverse analysis (BMFIA), which leverages simpler and computationally cheaper lower-fidelity (LF) models that are designed to provide model derivatives. BMFIA learns a simple, probabilistic dependence of the LF and HF models, which is then employed in an altered likelihood formulation to statistically correct the inaccurate LF response. From a Bayesian viewpoint, this dependence represents a multi-fidelity conditional density (discriminative model). We demonstrate how this multi-fidelity conditional density can be learned robustly in the small data regime from only a few HF and LF simulations (50 to 300), which would not be sufficient for naive surrogate approaches. The formulation is fully differentiable and allows the flexible design of a wide range of LF models.