The State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China.
HEDPS and LTCS, College of Engineering, Peking University, Beijing 100871, China.
Chaos. 2022 Dec;32(12):123134. doi: 10.1063/5.0114542.
A data-driven sparse identification method is developed to discover the underlying governing equations from noisy measurement data through the minimization of Multi-Step-Accumulation (MSA) in error. The method focuses on the multi-step model, while conventional sparse regression methods, such as the Sparse Identification of Nonlinear Dynamics method (SINDy), are one-step models. We adopt sparse representation and assume that the underlying equations involve only a small number of functions among possible candidates in a library. The new development in MSA is to use a multi-step model, i.e., predictions from an approximate evolution scheme based on initial points. Accordingly, the loss function comprises the total error at all time steps between the measured series and predicted series with the same initial point. This enables MSA to capture the dynamics directly from the noisy measurements, resisting the corruption of noise. By use of several numerical examples, we demonstrate the robustness and accuracy of the proposed MSA method, including a two-dimensional chaotic map, the logistic map, a two-dimensional damped oscillator, the Lorenz system, and a reduced order model of a self-sustaining process in turbulent shear flows. We also perform further studies under challenging conditions, such as noisy measurements, missing data, and large time step sizes. Furthermore, in order to resolve the difficulty of the nonlinear optimization, we suggest an adaptive training strategy, namely, by gradually increasing the length of time series for training. Higher prediction accuracy is achieved in an illustrative example of the chaotic map by the adaptive strategy.
一种基于数据驱动的稀疏辨识方法被开发出来，用于通过在误差中最小化多步累加（MSA）来从噪声测量数据中发现潜在的控制方程。该方法侧重于多步模型，而传统的稀疏回归方法，如非线性动力学稀疏辨识（SINDy）方法，则是单步模型。我们采用稀疏表示，并假设潜在的方程只涉及到库中可能候选函数中的少数几个函数。MSA 的新发展是使用多步模型，即基于初始点的近似演化方案的预测。因此，损失函数包括在相同初始点的测量系列和预测系列之间所有时间步的总误差。这使得 MSA 能够直接从噪声测量中捕捉动力学，抵制噪声的干扰。通过几个数值例子，我们展示了所提出的 MSA 方法的鲁棒性和准确性，包括二维混沌映射、逻辑映射、二维阻尼振荡器、Lorenz 系统以及湍流剪切流中自维持过程的降阶模型。我们还在具有挑战性的条件下进行了进一步的研究，例如噪声测量、数据缺失和大时间步长。此外，为了解决非线性优化的困难，我们提出了一种自适应训练策略，即通过逐渐增加训练的时间序列长度。在混沌映射的实例中，自适应策略实现了更高的预测精度。
