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

2025-05-30

AI Summary

High-dimensional (>100D) Bayesian inverse problems involving computationally expensive, nonsmooth, and analytically gradient-free high-fidelity (HF) multiphysics models face severe feasibility bottlenecks: adjoint derivatives are intractable to derive, and automatic differentiation is infeasible without extensive code refactoring. To address this, we propose the Bayesian Multi-Fidelity Inversion Algorithm (BMFIA), introducing— for the first time—a conditional density modeling-based multi-fidelity statistical correction scheme. BMFIA robustly learns low-/high-fidelity model dependencies using only 50–300 HF simulations, eliminating reliance on large-scale surrogate models. Integrating Bayesian inference, multi-fidelity modeling, and differentiable likelihood reconstruction, it leverages a differentiable low-fidelity model to provide gradients, fully circumventing differentiation of the HF code. We successfully solve a nonlinear transient poroelastic medium inversion problem with fine spatial resolution, demonstrating substantial improvements in computational efficiency and feasibility for high-dimensional stochastic inverse problems.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Probabilistic Inference
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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. We demonstrate that BMFIA solves Bayesian inverse problems for scenarios that used to be prohibitive, such as finely-resolved spatial reconstruction problems for nonlinear and transient coupled poro-elastic media physics.

Problem

• High-dimensional Bayesian inverse analysis for expensive, non-differentiable physics simulations
• Overcoming computational challenges in gradient-based inference for legacy multi-physics codes
• Enabling Bayesian inference with limited high-fidelity data via multi-fidelity modeling

Innovation

• Uses Bayesian multi-fidelity inverse analysis
• Leverages lower-fidelity models for derivatives
• Learns probabilistic HF-LF dependence efficiently