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

2026-03-04

AI Summary

This work proposes a unified framework that integrates hierarchical Bayesian inference with data-driven closure learning to address the inverse problem of model calibration in multiphysics systems, where unknown parameters and incomplete dynamical laws pose significant challenges. The approach jointly infers system-specific parameters and shared unknown dynamics across multiple related systems through a hierarchical structure. Neural networks—such as Fourier Neural Operators (FNOs) and parameterized Physics-Informed Neural Networks (PINNs)—are employed to construct closure models for ODEs/PDEs. Efficient posterior inference is achieved via maximum marginal likelihood estimation combined with ensemble Metropolis-adjusted Langevin algorithm (MALA) sampling. Furthermore, an adaptive surrogate forward model and a bilevel optimization strategy are introduced to substantially reduce the computational cost associated with repeated forward solves. Experiments demonstrate that the framework achieves high calibration accuracy while enabling efficient joint modeling and computational acceleration across systems.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

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).

Problem

• inverse problems
• closure learning
• hierarchical inference
• unknown dynamics
• model calibration

Innovation

• Hierarchical Bayesian Inference
• Closure Learning
• Surrogate Modeling
• Bilevel Optimization
• Neural Operators