Unrolled-SINDy: A Stable Explicit Method for Non linear PDE Discovery from Sparsely Sampled Data
This work addresses a key challenge in machine learning for physical dynamics modeling, particularly for real-world problems with sparse temporal sampling, representing an incremental improvement over existing SINDy approaches.
The paper tackles the problem of discovering governing differential equations from sparsely sampled time data, which existing SINDy-based methods often fail to address, and shows that the proposed Unrolled-SINDy method enables recovery of equation parameters not accessible to non-unrolled methods across various settings.
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.