Differential-algebraic equations (DAEs) integrate ordinary differential equations (ODEs) with algebraic constraints, providing a fundamental framework for developing models of dynamical systems characterized by timescale separation, conservation laws, and physical constraints. While sparse optimization has revolutionized model development by allowing data-driven discovery of parsimonious models from a library of possible equations, existing approaches for dynamical systems assume DAEs can be reduced to ODEs by eliminating variables before model discovery. This assumption limits the applicability of such methods for DAE systems with unknown constraints and time scales. We introduce Sparse Optimization for Differential-Algebraic Systems (SODAs), a data-driven method for the identification of DAEs in their explicit form. By discovering the algebraic and dynamic components sequentially without prior identification of the algebraic variables, this approach leads to a sequence of convex optimization problems. It has the advantage of discovering interpretable models that preserve the structure of the underlying physical system. To this end, SODAs improves numerical stability when handling high correlations between library terms, caused by near-perfect algebraic relationships, by iteratively refining the conditioning of the candidate library. We demonstrate the performance of our method on biological, mechanical, and electrical systems, showcasing its robustness to noise in both simulated time series and real-time experimental data.

翻译：微分代数方程（DAEs）将常微分方程（ODEs）与代数约束相结合，为建立具有时间尺度分离、守恒律和物理约束特性的动力系统模型提供了基础框架。尽管稀疏优化通过从候选方程库中数据驱动地发现简约模型，彻底改变了模型构建方式，但现有动力系统方法均假设DAEs可在模型发现前通过变量消元转化为ODEs。这一假设限制了此类方法在具有未知约束和时间尺度的DAE系统中的适用性。本文提出微分代数系统稀疏优化方法（SODAs），这是一种数据驱动的显式DAE辨识方法。该方法无需预先识别代数变量，通过顺序发现代数与动态分量，将问题转化为一系列凸优化问题。其优势在于能够发现保持底层物理系统结构、具有可解释性的模型。为此，SODAs通过迭代优化候选函数库的条件数，显著提升了处理因近似完美代数关系导致的函数项高相关性时的数值稳定性。我们在生物、机械和电气系统上验证了该方法的性能，展示了其对仿真时间序列和实时实验数据中噪声的鲁棒性。