Finite Difference Neural Networks: Fast Prediction of Partial Differential Equations
This work addresses the challenge of modeling complex systems in science and engineering, but it appears incremental as it builds on existing neural network and finite difference approaches.
The authors tackled the problem of learning partial differential equations from data by proposing a novel neural network framework called finite difference neural networks (FDNet), which achieved fast prediction of future dynamical behavior with few trainable parameters, as demonstrated on the heat equation with comparisons to the Forward Euler method.
Discovering the underlying behavior of complex systems is an important topic in many science and engineering disciplines. In this paper, we propose a novel neural network framework, finite difference neural networks (FDNet), to learn partial differential equations from data. Specifically, our proposed finite difference inspired network is designed to learn the underlying governing partial differential equations from trajectory data, and to iteratively estimate the future dynamical behavior using only a few trainable parameters. We illustrate the performance (predictive power) of our framework on the heat equation, with and without noise and/or forcing, and compare our results to the Forward Euler method. Moreover, we show the advantages of using a Hessian-Free Trust Region method to train the network.