Differential equation models have found applications to problems originating in disciplines such as biology, physics, chemistry, and medicine. In recent years, due to the abundance of data, there has been an active search for data-driven methods to learn Differential equation models from data. Advancements in neural networks and deep learning have motivated a shift towards data-driven deep learning methods that can learn differential equations from data. In this work, we propose a forward-Euler-based neural network model and test its performance by learning ODEs such as the FitzHugh-Nagumo equations from data using different numbers of hidden layers and different neural network widths. We also show the reliability of this approach at 0%, 1%, and 5% noise in the training data.
The paper is available at
https://arxiv.org/abs/2205.11483
- Formulate a neural network.
- A residual ODE connection inspired by the forward-Euler scheme will be used to construct the loss function.
- We demonstrate that a loss function built using a forward Euler temporal scheme in the network formulation can learn the form of an ODE from data to high accuracy.
Here, we show that training the network with simulated data, the network learns the dynamics of the dynamical system, and a predicted solution is obtained using the learned dynamics in an ODE solver such as scipy's odeint.
- Python 3.9.6
- import tensorflow.compat.v1 as tf
- Other packages, such as Matplotlib, Numpy and Scipy, are also used
The inspiration for this work is the following paper：
https://arxiv.org/abs/1801.01236