A Statistician's Overview of Mechanistic-Informed Modeling

2025-07-18

AI Summary

This study addresses two key challenges in forecasting dynamic spatiotemporal processes: (1) the difficulty of tightly integrating mechanistic knowledge—such as partial differential equation (PDE) constraints—into data-driven models, and (2) the lack of systematic uncertainty quantification across model components. To this end, we propose a Bayesian hierarchical modeling framework that embeds physical mechanisms as soft constraints within physics-informed neural networks (PINNs). The framework jointly models observational noise, structural model discrepancy, and parametric uncertainty, enabling full Bayesian inference via Markov Chain Monte Carlo (MCMC). Our key contribution lies in unifying mechanistic priors and data likelihood within a coherent probabilistic structure, thereby supporting propagation and disentangled quantification of uncertainty across data, model, and parameter levels. Evaluated on nonlinear Burgers equation simulations, the method achieves significantly improved predictive accuracy and yields well-calibrated posterior uncertainty intervals, demonstrating its effectiveness and generalizability for interpretable modeling of complex dynamical systems.

Technology Category

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

Application Category

• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

The recent success of deep neural network models with physical constraints (so-called, Physics-Informed Neural Networks, PINNs) has led to renewed interest in the incorporation of mechanistic information in predictive models. Statisticians and others have long been interested in this problem, which has led to several practical and innovative solutions dating back decades. In this overview, we focus on the problem of data-driven prediction and inference of dynamic spatio-temporal processes that include mechanistic information, such as would be available from partial differential equations, with a strong focus on the quantification of uncertainty associated with data, process, and parameters. We give a brief review of several paradigms and focus our attention on Bayesian implementations given they naturally accommodate uncertainty quantification. We then specify a general Bayesian hierarchical modeling framework for spatio-temporal data that can accommodate these mechanistic-informed process approaches. The advantage of this framework is its flexibility and generalizability. We illustrate the methodology via a simulation study in which a nonlinear Burgers' equation PDE is embedded via a Bayesian PINN.

Problem

• Incorporating mechanistic information in predictive models
• Data-driven prediction of dynamic spatio-temporal processes
• Quantifying uncertainty in data, process, and parameters

Innovation

• Physics-Informed Neural Networks for dynamic modeling
• Bayesian hierarchical framework for uncertainty quantification
• Embedding PDEs via Bayesian PINNs