NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data

TL;DR

NeuPDE introduces a neural network framework for modeling time-dependent data through ordinary and partial differential equations, effectively capturing dynamics and reducing parameter costs compared to traditional deep networks.

Contribution

This work presents a novel neural PDE-based method for data-driven modeling of dynamic systems, integrating differential structure with neural networks for improved accuracy and efficiency.

Findings

Successfully models various dynamical systems from data

Outperforms recurrent networks in data discovery tasks

Reduces parameter count on MNIST datasets

Abstract

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.