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.

翻译：数据采集与计算方法的进步加速了基于微分方程的复杂系统建模。此类系统通常由两个（或更多）耦合变量描述，但通常仅能获得其中一个变量的控制方程，而其余变量只能通过数据获取。已知物理规律与观测数据之间的这种不匹配对现有物理信息机器学习方法构成了根本性挑战，这些方法通常假设要么完全掌握控制方程，要么所有变量均有完整数据可用。本文提出MUSIC（稀疏与不完整约束下的多任务学习），一种稀疏性诱导的多任务神经网络框架，通过将部分物理约束与数据驱动学习相结合，在物理约束变量与数据驱动变量互斥的情况下恢复耦合系统的全维度解。MUSIC采用无网格（随机）训练数据采样与稀疏正则化技术，生成高度压缩的模型，显著提升了训练与评估效率。我们证明，在数据稀缺和噪声条件下，MUSIC能准确学习复杂耦合系统的解（冲击波解、间断解、模式形成解），其性能始终优于非稀疏模型。这些结果表明，MUSIC为物理知识不完整的部分观测系统建模提供了一种灵活有效的解决方案。