Seq-SVF: An unsupervised data-driven method for automatically identifying hidden governing equations

Abstract

In this work, an unsupervised data-driven method based on sequential singular value filtering (Seq-SVF) is proposed to simultaneously identify multiple partial differential equations from observed data considering potential noises. This method is aimed to extend the Sparse Identification of Nonlinear Dynamics (SINDy) to the identification of general nonlinear partial differential equations by transforming the paradigm based on regression to an unsupervised paradigm. To discover the complex coupled equations of vector or tensor forms without prior knowledge, the techniques of singular value decomposition (SVD) and strong rank-revealing QR factorization (sRRQR) are applied to the data matrix, which ensures that the method can automatically identify the number and the corresponding linearly independent terms as the left-hand terms of governing equations. To balance the complexity and the precision of modeling, a strategy for filtering singular values is designed to determine the sparse structure of governing equations from a large number of nonlinear basis functions. We show the success of the method to extract explicit and succinct models from many complex linear, nonlinear, and multiphysics mechanical systems, and the examples show more accuracy compared with using traditional sparse learning methods.