Discovering nonlinear differential equations that describe system dynamics from empirical data is a fundamental challenge in contemporary science. Here, we propose a methodology to identify dynamical laws by integrating denoising techniques to smooth the signal, sparse regression to identify the relevant parameters, and bootstrap confidence intervals to quantify the uncertainty of the estimates. We evaluate our method on well-known ordinary differential equations with an ensemble of random initial conditions, time series of increasing length, and varying signal-to-noise ratios. Our algorithm consistently identifies three-dimensional systems, given moderately-sized time series and high levels of signal quality relative to background noise. By accurately discovering dynamical systems automatically, our methodology has the potential to impact the understanding of complex systems, especially in fields where data are abundant, but developing mathematical models demands considerable effort.

翻译：从经验数据中发现描述系统动力学的非线性微分方程是当代科学中的一个基本挑战。本文提出了一种通过整合去噪技术平滑信号、稀疏回归识别相关参数以及自助法置信区间量化估计不确定性的方法，来识别动力学规律。我们在具有随机初始条件集合、长度递增的时间序列以及不同信噪比的已知常微分方程上评估了该方法。给定中等规模的时间序列和相对于背景噪声足够高的信号质量，我们的算法能够持续识别出三维系统。通过自动准确发现动力学系统，该方法有望影响对复杂系统的理解，尤其是在数据丰富但建立数学模型需要大量努力的领域中。