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

TL;DR

This paper explores novel loss functions for artificial neural networks to effectively solve ordinary differential equations, ensuring the neural network solutions satisfy both the equations and their initial or boundary conditions.

Contribution

It introduces a generalized loss function for nth-order ODEs and evaluates various construction methods across multiple models.

Findings

Effective loss functions for nth-order ODEs

Improved neural network solutions satisfying boundary conditions

Assessment of different construction methods

Abstract

This paper investigates the use of artificial neural networks (ANNs) to solve differential equations (DEs) and the construction of the loss function which meets both differential equation and its initial/boundary condition of a certain DE. In section 2, the loss function is generalized to $n^\text{th}$ order ordinary differential equation(ODE). Other methods of construction are examined in Section 3 and applied to three different models to assess their effectiveness.