The objective of the current study is to analyze the neural network solution for different types of differential equations such as initial value, boundary values, Dirichlet, and Neumann boundary conditions. These systems of equations are of critical value in engineering applications. The trained neural network on the physics-based governing equations do not depend on the discretization and could be deployed for real-time analysis. The neural network solutions are in a closed form, completely differentiable and could be used for further computation analysis. In this study backward automatic differentiation was used for training the neural network. The novelty of this work was training neural networks without any target data and the solution thus formed is a part of self-supervised algorithms.
This work is inspired by the work of Lagaris et al.
Different differential equations have been solved to demonstrate the effectivness of the neural network to solve the mathematical equations without using any kind of training data. The validation of the proposed methods was done with help of the analytical solution available.
(a) Initial Value The mathematical equation to be solved is :
(b) Boundary Value
(c) Coupled Equation
(d) Dirichlet Boundary Condition
(e) Neumann Boundary Condition