Learning chaotic systems from noisy data via multi-step optimization and adaptive training

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.