Data-driven derivation of partial differential equations using neural network model

It has become critical to develop explanatory models using partial differential equations (PDEs) for big data from interesting scientific phenomena, e.g., fluids, atmosphere, and cosmic phenomena. Big data are often measured at discrete spatiotemporal points. In this paper, we assume a PDE and consider a set of the PDE solution data at multiple discrete points as pseudo-measurement data. In a data-driven PDE derivation, it is assumed that the PDE is a linear regression model comprising partial temporal and spatial differential terms, where the coefficients are estimated using regression analysis techniques, and the PDE derivation accuracy is defined as the difference (i.e., error) between the exact and estimated coefficients. A spatiotemporal model is required to calculate the partial temporal and spatial differential terms; thus, we employ a spatiotemporal model based on a neural network (NN) obtained from big data. To develop the data-driven PDE derivation technique, we employ PDEs with exact solutions such that we can calculate the differential terms by differentiating them analytically at multiple discrete points. If we assume an activation function in the NN model, the partial differential term can be derived using a chain rule. The NN model accuracy is measured by the loss function and error between the exact and estimated partial differential terms. In addition, we clarify a requirement for an NN structure that can maximize the PDE derivation and NN model accuracy by varying the NN meta-parameters, i.e., the numbers of NN layers and neurons.