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).

翻译：反问题是通过校准模型以匹配数据的任务。它们在不同工程应用中发挥着关键作用，使从业者能够将模型与现实对齐。在许多应用中，工程师和科学家无法完整掌握：i) 系统的详细属性（如材料特性、几何结构、初始条件等）；ii) 描述所有作用动力学的完整定律（如摩擦定律、复杂阻尼现象及一般非线性相互作用）。本文提出一种原理性方法，利用来自不同但相关物理系统集合的数据，共同估计每个系统的个体模型参数，并以基于机器学习的闭合模型形式学习共享的未知动力学。为稳健推断每个系统的未知参数，我们采用分层贝叶斯框架，该框架支持对多个系统及其总体层面统计量的联合推断。为学习闭合项，我们使用嵌入在问题常微分方程/偏微分方程表述中的神经网络的最大边缘似然估计。为实现该框架，我们采用集成Metropolis-Adjusted Langevin Algorithm (MALA) 以进行稳定高效的采样。为缓解求解反问题时重复正向评估的计算瓶颈，我们引入双层优化策略，在推理过程中同步训练代理正向模型。在此框架内，我们评估并比较了不同的代理架构，特别是傅里叶神经算子 (FNO) 与参数化物理信息神经网络 (PINN)。