# Learning Coupled System Dynamics under Incomplete Physical Constraints and Missing Data

arXiv: https://arxiv.org/abs/2512.23761 · HTML: https://mlsanity.com/paper/2512.23761/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 43 references that are in the map.
- First posted on arXiv 28 Dec 2025.

## Builds on (most cited first, 12 of 12)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Learning Sparse Neural Networks through L0 Regularization](https://mlsanity.com/paper/1712.01312/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [AI-Aristotle: A Physics-Informed framework for Systems Biology Gray-Box Identification](https://mlsanity.com/paper/2310.01433/) (2023)
- [PDE-LEARN: Using Deep Learning to Discover Partial Differential Equations from Noisy, Limited Data](https://mlsanity.com/paper/2212.04971/) (2022)
- [Learning Dynamical Systems with Side Information](https://mlsanity.com/paper/2008.10135/) (2020)
- [SPADE4: Sparsity and Delay Embedding based Forecasting of Epidemics](https://mlsanity.com/paper/2211.08277/) (2022)
- [Dispersion based Recurrent Neural Network Model for Methane Monitoring in Albertan Tailings Ponds](https://mlsanity.com/paper/2411.06741/) (2024)
- [Unified Spatiotemporal Physics-Informed Learning (USPIL): A Framework for Modeling Complex Predator-Prey Dynamics](https://mlsanity.com/paper/2509.13425/) (2025)

## Built on by (most cited first, 2 of 2)

- [Learning PDE Time-Stepping with Neural Cellular Automata](https://mlsanity.com/paper/2608.30328/) (2026)
- [Random Forest-Informed Cellular Automaton for Large-Scale Wildfire Spread Modelling](https://mlsanity.com/paper/2609.01675/) (2026)

## Abstract (arXiv)

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.