Unrolled-SINDy: A Stable Explicit Method for Non linear PDE Discovery from Sparsely Sampled Data

2025-10-21

AI Summary

This study addresses the numerical instability of physics-informed PDE identification via Sparse Identification of Nonlinear Dynamics (SINDy) under sparse temporal sampling. We propose Unrolled-SINDy, a framework that explicitly unrolls numerical integration schemes (e.g., Euler or RK4), thereby decoupling the model’s internal time step from the irregular or coarse observation sampling rate. This design mitigates the ill-conditioning and error accumulation inherent in conventional SINDy under sparse sampling. The method integrates closed-form iterative updates with gradient-based optimization and incorporates an unrolling mechanism to enhance robustness. Extensive experiments under multi-level noise demonstrate substantial improvements in parameter recovery accuracy and equation discovery capability. Notably, Unrolled-SINDy successfully reconstructs nonlinear dynamical structures—such as quadratic and cubic terms—that standard SINDy fails to identify under sparse sampling. Overall, the approach significantly extends the applicability boundary of data-driven modeling for physical systems operating under severely limited observational data.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Identifying from observation data the governing differential equations of a physical dynamics is a key challenge in machine learning. Although approaches based on SINDy have shown great promise in this area, they still fail to address a whole class of real world problems where the data is sparsely sampled in time. In this article, we introduce Unrolled-SINDy, a simple methodology that leverages an unrolling scheme to improve the stability of explicit methods for PDE discovery. By decorrelating the numerical time step size from the sampling rate of the available data, our approach enables the recovery of equation parameters that would not be the minimizers of the original SINDy optimization problem due to large local truncation errors. Our method can be exploited either through an iterative closed-form approach or by a gradient descent scheme. Experiments show the versatility of our method. On both traditional SINDy and state-of-the-art noise-robust iNeuralSINDy, with different numerical schemes (Euler, RK4), our proposed unrolling scheme allows to tackle problems not accessible to non-unrolled methods.

Problem

• Discovering nonlinear PDEs from sparsely sampled data
• Improving stability of explicit methods for PDE identification
• Decoupling numerical time step from data sampling rate

Innovation

• Unrolled-SINDy uses unrolling scheme for stability
• Decouples numerical time step from data sampling rate
• Enables gradient descent or closed-form solution approaches