Learning Coupled System Dynamics under Incomplete Physical Constraints and Missing Data

TL;DR

This paper introduces MUSIC, a neural network framework that combines partial physical constraints with data-driven learning to accurately model complex coupled systems with incomplete knowledge and limited data.

Contribution

MUSIC is a novel multitask neural network that integrates sparse physical constraints with data, enabling full solution recovery in coupled systems with incomplete physics and data scarcity.

Findings

MUSIC accurately models shock waves and pattern formation.

It outperforms non-sparse models under noisy, data-scarce conditions.

The approach is mesh-free and highly efficient.

Abstract

Advances in data acquisition and computational methods have accelerated the use of differential equation based modelling for complex systems. Such systems are often described by coupled (or more) variables, yet governing equation is typically available for one variable, while the remaining variable can be accessed only through data. This mismatch between known physics and observed data poses a fundamental challenge for existing physics-informed machine learning approaches, which generally assume either complete knowledge of the governing equations or full data availability across all variables. In this paper, we introduce MUSIC (Multitask Learning Under Sparse and Incomplete Constraints), a sparsity induced multitask neural network framework that integrates partial physical constraints with data-driven learning to recover full-dimensional solutions of coupled systems when physics-constrained and data-informed variables are mutually exclusive. MUSIC employs mesh-free (random) sampling of training data and sparsity regularization, yielding highly compressed models with improved training and evaluation efficiency. We demonstrate that MUSIC accurately learns solutions (shock wave solutions, discontinuous solutions, pattern formation solutions) to complex coupled systems under data-scarce and noisy conditions, consistently outperforming non-sparse formulations. These results highlight MUSIC as a flexible and effective approach for modeling partially observed systems with incomplete physical knowledge.