Learning Coupled System Dynamics under Incomplete Physical Constraints and Missing Data

TL;DR

The paper addresses learning full-field solutions of coupled PDEs when physics is known only for some variables and data are available only for others. It introduces MUSIC, a sparse multitask neural network that enforces data-fitting for data variables and PDE residuals for equation variables, with ℓ0-based sparsity and mesh-free sampling to enable training under disjoint priors. The method minimizes a loss L that blends data fitting terms and PDE residuals, plus an ℓ0 sparsity penalty, e.g., L = λ1 ||u1-û1||^2 + ... + λn ||∂ûn/∂t - Fn(...)||^2 + Λ||Θ||_0, and uses hard-thresholding or hard-concrete relaxations for compression. The approach is demonstrated on SWE, FN, λ−ω RD, and a wildfire model, showing accurate recovery of full-field solutions under data scarcity and noise, along with substantial model compression. Compared to PINN-based approaches, MUSIC better handles disjoint priors, enables data-efficient learning, and offers robust performance across diverse spatiotemporal dynamics.

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.

Figures (15)

• Figure 1: True solutions $\{h,hu\}$ of the dam break problem modeled by SWE at $t=\{0,0.2,0.4,0.6,0.8,1.0\}$.
• Figure 2: True and learned solutions $h$ and $hu$ using a 4 layer model with 20 neurons with $N_s=100$ and $N_t=800$. Top row: True and learned solution for equation variable $h(x,t)$. Bottom row: True and learned solution for data variable $(hu)(x,t)$.
• Figure 3: Dataset creation for the FN system. The red dots illustrate the evenly spaced points chosen in the spatial domain. The full dataset is built by taking 50 evenly spaced temporal solutions i.e., solutions at $t=\{10,11,12,\dots,59,60\}$.
• Figure 4: Solution to the FH system with random initialization learned using a 4 layer fully connected neural networks using 200 neurons with $N_{s_x}*N_{s_y} = 5000$ and $N_t = 40$. Note the scales for the error are different for each plot.
• Figure 5: Error evolution of learning solutions to the FN system across different timesteps and noise levels. $N_{s_x}*N_{s_y} = 5000$ and $N_t = 40$.