Supplementary Material from Sparse identification of nonlinear dynamics from under-sampled periodic datasets

Accurate numerical differentiation is crucial when preprocessing discrete data to enable discovery of underlying governing equations in a dynamical system, such as that required in implementing the Sparse Identification of Nonlinear Dynamics (SINDy) approach. This is particularly important due to the sensitivity of finite-difference schemes to noise. Two common denoising techniques are polynomial interpolation for spatial derivatives and total variation regularization for temporal derivatives. However, these methods are ineffective for under-sampled datasets. This paper introduces a novel approach, termed data folding, to overcome this limitation for sparse periodic datasets. The method enhances numerical differentiation accuracy in proportion to the dataset size and phase accuracy and also reduces noise in the matrix of candidate functions. The effectiveness of data folding is demonstrated with four different under-sampled datasets: two essentially noiseless, artificially generated solutions of the Lorenz equations under periodic and double-periodic conditions, and two datasets from experiments involving acoustically forced combustion instabilities with low and high noise levels. In all cases, the numerical model generated by SINDy accurately reproduced the correct temporal dynamics only when data folding was applied beforehand.