Efficient Closure Learning in Dynamical Systems: Embedded vs. Staged Data-Driven Approaches Compared

Forum topic · 小凯 · 2026-11-03

Summary

This paper (arXiv 2611.03612, by Rileigh Bandy, Teresa Portone, Mohammad Khalil, Eric Crislip, and Kyle Neal) compares embedded versus staged approaches to data-driven closure-model learning in dynamical systems. Closure models approximate unresolved effects in reduced or incomplete dynamical systems, enabling prediction of phenomena beyond direct simulation capability. Embedded methods, which train the closure inside the computational model, achieve high accuracy but are computationally expensive and can cause numerical instability during training. Staged methods instead use data assimilation to infer state and closure trajectories, then train the machine-learning closure offline, avoiding repeated simulator calls. Using a synthetic SIQR epidemiology model with nonlinear state-dependent quarantine closure, the authors characterize the cost-accuracy trade-off: embedded training delivers the strongest average accuracy, while staged methods achieve comparable qualitative behavior with modestly increased error but require significantly fewer model evaluations and offer better optimization robustness. The results provide practical guidance for choosing closure-learning strategies in computational science workflows.

Paper Overview

Field: Machine Learning
Authors: Rileigh Bandy, Teresa Portone, Mohammad Khalil, Eric Crislip, Kyle Neal
Published: 2026-11-02
arXiv: 2611.03612

Abstract

Increased computing power has transformed computational science and improved predictive capabilities for complex systems, but many phenomena still exceed current simulation capabilities or require rapid evaluations for fast design iterations. Closure models address this challenge by approximating unresolved effects in reduced or incomplete dynamical systems.

Data-driven closures offer a flexible mechanism for improving model accuracy and applicability, but approaches that embed the closure within a computational model for construction and training phases can be computationally expensive and cause numerical instabilities in the computational model during training. To avoid the challenges posed by embedded approaches, staged approaches to closure-model learning have been developed: the first stage uses data assimilation to infer state and closure trajectories from observations; the second stage trains a machine-learning closure model offline on the assimilated data, avoiding repeated simulator calls during training.

This work examines the cost-accuracy trade-off between embedded and staged data-driven closure modeling using a synthetic SIQR epidemiology model with a nonlinear state-dependent quarantine closure as a test case.

Key Findings

The study demonstrates a practical accuracy-efficiency trade-off:
Embedded training provides the strongest average accuracy.
Staged methods trade modestly increased error for significantly fewer model evaluations and better optimization robustness, with comparable qualitative behavior.

Discussion

The results suggest that staged closure learning is an attractive option when computational budget or training stability is a concern, while embedded training remains preferable when maximum accuracy is required and the cost can be afforded.

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