Dynamical systems are typically governed by a set of linear/nonlinear differential equations. Distilling the analytical form of these equations from very limited data remains intractable in many disciplines such as physics, biology, climate science, engineering and social science. To address this fundamental challenge, we propose a novel Physics-informed Spline Learning (PiSL) framework to discover parsimonious governing equations for nonlinear dynamics, based on sparsely sampled noisy data. The key concept is to (1) leverage splines to interpolate locally the dynamics, perform analytical differentiation and build the library of candidate terms, (2) employ sparse representation of the governing equations, and (3) use the physics residual in turn to inform the spline learning. The synergy between splines and discovered underlying physics leads to the robust capacity of dealing with high-level data scarcity and noise. A hybrid sparsity-promoting alternating direction optimization strategy is developed for systematically pruning the sparse coefficients that form the structure and explicit expression of the governing equations. The efficacy and superiority of the proposed method have been demonstrated by multiple well-known nonlinear dynamical systems, in comparison with two state-of-the-art methods.

翻译：典型的动态系统由一组线性/非线性差异方程式管理。从非常有限的数据中提炼这些方程式的分析形式在许多学科,例如物理学、生物学、气候科学、工程学和社会科学中仍然难以找到。为了应对这一根本挑战,我们提议了一个全新的物理知情的Spline Learning(PISL)框架,以发现非线性动态的模糊的治理方程式,其基础是稀有抽样的吵闹数据。关键概念是:(1) 利用样条来对动态进行本地内插,进行分析性区分,建立候选术语库;(2) 使用极少的正方程式代表,(3) 利用物理残留物作为样条学习的参考; 样条和发现的基本物理学之间的协同作用导致处理高水平数据稀缺和噪音的强大能力; 开发了一种混合的松动-促进交替方向优化战略,以便系统地处理构成正统方程式的结构和明确表达的稀薄系数。 与两种州级方法相比,多种已知的非线性动态系统证明了拟议方法的功效和优越性。