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.

翻译：从观测数据中识别物理动力学所遵循的微分方程是机器学习中的一个关键挑战。尽管基于SINDy的方法在这一领域已展现出巨大潜力，但它们仍无法解决一整类现实世界问题，即数据在时间上稀疏采样的情形。本文介绍了Unrolled-SINDy，这是一种利用展开方案来提高偏微分方程发现中显式方法稳定性的简单方法。通过将数值时间步长与可用数据的采样率解耦，我们的方法能够恢复那些由于局部截断误差过大而无法成为原始SINDy优化问题最小化解的方程参数。我们的方法既可通过迭代闭式方法实现，也可通过梯度下降方案实现。实验展示了我们方法的通用性。无论是在传统的SINDy上，还是在最先进的抗噪iNeuralSINDy上，结合不同的数值格式（欧拉法、RK4法），我们提出的展开方案都能处理非展开方法无法解决的问题。