NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data
This method addresses the challenge of modeling dynamic data for researchers in computational science and machine learning, but it appears incremental as it builds on existing data-discovery techniques.
The paper tackles the problem of extracting governing differential equations from time-dependent data by proposing a neural network-based approach that parameterizes unknown models with multilayer perceptrons and nonlinear differential terms, demonstrating it on dynamical systems and showing reduced parameter costs on MNIST and Fashion MNIST datasets.
We propose a neural network based approach for extracting models from dynamic data using ordinary and partial differential equations. In particular, given a time-series or spatio-temporal dataset, we seek to identify an accurate governing system which respects the intrinsic differential structure. The unknown governing model is parameterized by using both (shallow) multilayer perceptrons and nonlinear differential terms, in order to incorporate relevant correlations between spatio-temporal samples. We demonstrate the approach on several examples where the data is sampled from various dynamical systems and give a comparison to recurrent networks and other data-discovery methods. In addition, we show that for MNIST and Fashion MNIST, our approach lowers the parameter cost as compared to other deep neural networks.