Stabilizing PDE--ML Coupled System

2025-06-23

AI Summary

Long-term numerical integration of PDE–machine learning (ML) coupled systems frequently suffers from instability, particularly due to the inability of surrogate models to represent unresolved-scale dynamics. Method: Focusing on the viscous Burgers equation as a prototypical example, this work identifies the root cause of instability and proposes a stabilization strategy grounded in the Mori–Zwanzig (M–Z) formalism. By explicitly incorporating memory terms that model decoherence effects, the approach corrects dynamic biases in the surrogate model without requiring higher-fidelity training or hard physical constraints. Contribution/Results: The M–Z correction significantly improves numerical stability and long-term predictive accuracy of time integration. Experiments demonstrate effective suppression of canonical instability modes. This work establishes a new paradigm for robust, data-driven modeling of complex multiscale physical systems.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Multiagent Systems: Multiagent Learning

Application Category

• User Modeling, Personalization and Recommendation: Studies of user behavior, including longitudinal effects of personalized systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

A long-standing obstacle in the use of machine-learnt surrogates with larger PDE systems is the onset of instabilities when solved numerically. Efforts towards ameliorating these have mostly concentrated on improving the accuracy of the surrogates or imbuing them with additional structure, and have garnered limited success. In this article, we study a prototype problem and draw insights that can help with more complex systems. In particular, we focus on a viscous Burgers'-ML system and, after identifying the cause of the instabilities, prescribe strategies to stabilize the coupled system. To improve the accuracy of the stabilized system, we next explore methods based on the Mori--Zwanzig formalism.

Problem

• Address instabilities in PDE-ML coupled systems
• Stabilize viscous Burgers'-ML system dynamics
• Enhance accuracy using Mori-Zwanzig formalism

Innovation

• Stabilizing coupled PDE-ML systems
• Using viscous Burgers'-ML prototype
• Applying Mori-Zwanzig formalism accuracy