Recent advances in the field of data-driven dynamics allow for the discovery of ODE systems using state measurements. One approach, known as Sparse Identification of Nonlinear Dynamics (SINDy), assumes the dynamics are sparse within a predetermined basis in the states and finds the expansion coefficients through linear regression with sparsity constraints. This approach requires an accurate estimation of the state time derivatives, which is not necessarily possible in the high-noise regime without additional constraints. We present an approach called Derivative-based SINDy (DSINDy) that combines two novel methods to improve ODE recovery at high-noise levels. First, we denoise the state variables by applying a projection operator that leverages the assumed basis for the system dynamics. Second, we use a second order cone program (SOCP) to find the derivative and governing equations simultaneously. We derive theoretical results for the projection-based denoising step, which allow us to estimate the values of hyperparameters used in the SOCP formulation. This underlying theory helps limit the number of required user-specified parameters. We present results demonstrating that our approach leads to improved system recovery for the Van der Pol oscillator, the Duffing oscillator, the Rössler attractor, and the Lorenz 96 model.

数据驱动动力学领域的最新进展允许使用状态测量发现 ODE 系统。一种称为非线性动力学稀疏识别 (SINDy) 的方法假设动力学在状态中的预定基础内是稀疏的，并通过具有稀疏约束的线性回归找到扩展系数。这种方法需要准确估计状态时间导数，这在没有额外约束的情况下在高噪声状态下不一定可行。我们提出了一种称为基于导数的 SINDy (DSINDy) 的方法，它结合了两种新颖的方法来提高高噪声水平下的 ODE 恢复。首先，我们通过应用利用系统动力学假设基础的投影算子来对状态变量进行去噪。第二，我们使用二阶锥程序 (SOCP) 同时求导数和控制方程。我们得出了基于投影的去噪步骤的理论结果，这使我们能够估计 SOCP 公式中使用的超参数值。这一基本理论有助于限制所需的用户指定参数的数量。我们展示的结果表明，我们的方法可以改善 Van der Pol 振荡器、Duffing 振荡器、Rössler 吸引子和 Lorenz 96 模型的系统恢复。