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

arXiv: https://arxiv.org/abs/1812.08625 · HTML: https://mlsanity.com/paper/1812.08625/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Dec 2018.

## Built on by (most cited first, 17 of 17)

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- [Exact Constraint Enforcement in Physics-Informed Extreme Learning Machines using Null-Space Projection Framework](https://mlsanity.com/paper/2601.10999/) (2026)
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- [Comparative Study of Bending Analysis using Physics-Informed Neural Networks and Numerical Dynamic Deflection in Perforated nanobeam](https://mlsanity.com/paper/2604.24768/) (2026)
- [Effective Dimensionality as an Operator Invariant for Physics-Preserving Constraint Adaptation in Physics-Informed Neural Networks](https://mlsanity.com/paper/2606.06171/) (2026)

## Abstract (arXiv)

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.