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

## TL;DR

This paper introduces deep Theory of Functional Connections (TFC), a novel neural network-based method that transforms PDE boundary value problems into unconstrained optimization tasks, enabling grid-free, analytical solutions.

## Contribution

The work combines neural networks with TFC to solve PDEs without domain discretization, providing a closed-form, differentiable solution across the entire domain.

## Key findings

- Successfully applied to four PDE problems with various boundary conditions.
- Produces a continuous, analytical approximation of solutions after training.
- Avoids grid-based discretization and interpolation used in traditional methods.

## Abstract

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. This methodology has two major differences when compared with popular methods used to estimate the solutions of PDEs. First, this methodology does not need to discretize the domain into a grid, rather, this methodology randomly samples points from the domain during the training phase. Second, after training, this methodology represents a closed form, analytical, differentiable approximation of the solution throughout the entire training domain. In contrast, other popular methods require interpolation if the estimated solution is desired at points that do not lie on the discretized grid. The deep TFC method for estimating the solution of PDEs is demonstrated on four problems with a variety of boundary conditions.