Deep Theory of Functional Connections: A New Method for Estimating the Solutions of PDEs

Understand

This article presents a new methodology called deep Theory of Functional Connections (TFC) that estimates the solutions of partial differential equations (PDEs) by combining neural networks with TFC.

• TFC is used to transform PDEs with boundary conditions into unconstrained optimization problems by embedding the boundary conditions into a "constrained expression." In this work, a neural network is chosen as the free function, and used to solve the now unconstrained optimization problem.
• The loss function is taken as the square of the residual of the PDE.
• Then, the neural network is trained in an unsupervised manner to solve the unconstrained optimization problem.