ODEN: A Framework to Solve Ordinary Differential Equations using Artificial Neural Networks

05/28/2020
by   Liam L. H. Lau, et al.
0

We explore in detail a method to solve ordinary differential equations using feedforward neural networks. We prove a specific loss function, which does not require knowledge of the exact solution, to be a suitable standard metric to evaluate neural networks' performance. Neural networks are shown to be proficient at approximating continuous solutions within their training domains. We illustrate neural networks' ability to outperform traditional standard numerical techniques. Training is thoroughly examined and three universal phases are found: (i) a prior tangent adjustment, (ii) a curvature fitting, and (iii) a fine-tuning stage. The main limitation of the method is the nontrivial task of finding the appropriate neural network architecture and the choice of neural network hyperparameters for efficient optimization. However, we observe an optimal architecture that matches the complexity of the differential equation. A user-friendly and adaptable open-source code (ODEN) is provided on GitHub.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/29/2022

New Designed Loss Functions to Solve Ordinary Differential Equations with Artificial Neural Network

This paper investigates the use of artificial neural networks (ANNs) to ...
research
10/14/2022

Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations

In this work, we assess the ability of physics-informed neural networks ...
research
08/07/2022

Stochastic Scaling in Loss Functions for Physics-Informed Neural Networks

Differential equations are used in a wide variety of disciplines, descri...
research
06/06/2011

Constructing Runge-Kutta Methods with the Use of Artificial Neural Networks

A methodology that can generate the optimal coefficients of a numerical ...
research
01/12/2023

Universality of neural dynamics on complex networks

This paper discusses the capacity of graph neural networks to learn the ...
research
11/14/2020

Discovery of the Hidden State in Ionic Models Using a Domain-Specific Recurrent Neural Network

Ionic models, the set of ordinary differential equations (ODEs) describi...
research
12/03/2020

Computational characteristics of feedforward neural networks for solving a stiff differential equation

Feedforward neural networks offer a promising approach for solving diffe...

Please sign up or login with your details

Forgot password? Click here to reset