针对稀疏时间数据，提出可稳定发现非线性偏微分方程的新方法。
Unrolled-SINDy: A Stable Explicit Method for Non linear PDE Discovery from Sparsely Sampled Data
- 通过展开机制解耦时间步长与采样率，提升显式方法稳定性。
- 在稀疏数据下恢复出原SINDy优化无法获取的方程参数。
- 适用于传统SINDy和抗噪iNeuralSINDy，兼容多种数值格式。
从观测数据中识别物理系统所遵循的微分方程是机器学习中的关键挑战。尽管基于SINDy的方法在此领域展现出巨大潜力，但仍难以处理时间上稀疏采样的真实世界问题。本文提出Unrolled-SINDy，一种基于展开机制的简单方法，显著提升了显式方法在偏微分方程发现中的稳定性。通过将数值时间步长与可用数据采样率解耦，该方法可恢复因局部截断误差过大而无法被原始SINDy优化问题求得的方程参数。本方法可通过迭代闭式求解或梯度下降实现。实验表明其通用性强：在传统SINDy及前沿的噪声鲁棒iNeuralSINDy上，结合欧拉法与四阶龙格-库塔法（RK4），所提展开方案均能解决非展开方法无法处理的问题。
原文摘要 · Abstract (English)
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.
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