Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEs

TL;DR

This paper introduces a hierarchical Bayesian approach combined with neural network surrogates to jointly estimate parameters and learn unknown dynamics in inverse problems for ODEs and PDEs, improving efficiency and robustness.

Contribution

It develops a novel framework integrating hierarchical Bayesian inference with neural surrogates and bilevel optimization for inverse problems involving complex physical systems.

Findings

Hierarchical Bayesian inference effectively estimates system parameters.

Neural surrogates reduce computational costs in inverse problem solving.

FNO and PINNs are evaluated as surrogate models for efficiency.

Abstract

Inverse problems are the task of calibrating models to match data. They play a pivotal role in diverse engineering applications by allowing practitioners to align models with reality. In many applications, engineers and scientists do not have a complete picture of i) the detailed properties of a system (such as material properties, geometry, initial conditions, etc.); ii) the complete laws describing all dynamics at play (such as friction laws, complicated damping phenomena, and general nonlinear interactions). In this paper, we develop a principled methodology for leveraging data from collections of distinct yet related physical systems to jointly estimate the individual model parameters of each system, and learn the shared unknown dynamics in the form of an ML-based closure model. To robustly infer the unknown parameters for each system, we employ a hierarchical Bayesian framework, which allows for the joint inference of multiple systems and their population-level statistics. To learn the closures, we use a maximum marginal likelihood estimate of a neural network embeded within the ODE/PDE formulation of the problem. To realize this framework we utilize the ensemble Metropolis-Adjusted Langevin Algorithm (MALA) for stable and efficient sampling. To mitigate the computational bottleneck of repetitive forward evaluations in solving inverse problems, we introduce a bilevel optimization strategy to simultaneously train a surrogate forward model alongside the inference. Within this framework, we evaluate and compare distinct surrogate architectures, specifically Fourier Neural Operators (FNO) and parametric Physics-Informed Neural Network (PINNs).