Governing equation discovery based on causal graph for nonlinear dynamic systems

Abstract

Abstract The governing equations of nonlinear dynamic systems is of great significance for understanding the internal physical characteristics. In order to learn the governing equations of nonlinear systems from noisy observed data, we propose a novel method named governing equation discovery based on causal graph that combines spatio-temporal graph convolution network with governing equation modeling. The essence of our method is to first devise the causal graph encoding based on transfer entropy to obtain the adjacency matrix with causal significance between variables. Then, the spatio-temporal graph convolutional network is used to obtain approximate solutions for the system variables. On this basis, automatic differentiation is applied to obtain basic derivatives and form a dictionary of candidate algebraic terms. Finally, sparse regression is used to obtain the coefficient matrix and determine the explicit formulation of the governing equations. We also design a novel cross-combinatorial optimization strategy to learn the heterogeneous parameters that include neural network parameters and control equation coefficients. We conduct extensive experiments on seven datasets from different physical fields. The experimental results demonstrate the proposed method can automatically discover the underlying governing equation of the systems, and has great robustness.