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Deep Theory of Functional Connections: A New Method for Estimating the Solutions of Partial Differential Equations

Aerospace Engineering, Texas A&M University, College Station, TX 77843, USA
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Mach. Learn. Knowl. Extr. 2020, 2(1), 37-55; https://doi.org/10.3390/make2010004
Received: 11 February 2020 / Revised: 8 March 2020 / Accepted: 9 March 2020 / Published: 12 March 2020
(This article belongs to the Section Learning)
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 the TFC. The TFC is used to transform PDEs into unconstrained optimization problems by analytically embedding the PDE’s constraints into a “constrained expression” containing a free function. In this research, the free function is chosen to be a neural network, which is used to solve the now unconstrained optimization problem. This optimization problem consists of minimizing a loss function that is chosen to be the square of the residuals of the PDE. The neural network is trained in an unsupervised manner to minimize this loss function. 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 can randomly sample points from the domain during the training phase. Second, after training, this methodology produces an accurate analytical approximation of the solution throughout the entire training domain. Because the methodology produces an analytical solution, it is straightforward to obtain the solution at any point within the domain and to perform further manipulation if needed, such as differentiation. In contrast, other popular methods require extra numerical techniques if the estimated solution is desired at points that do not lie on the discretized grid, or if further manipulation to the estimated solution must be performed. View Full-Text
Keywords: deep learning; neural network; theory of functional connections; partial differential equation deep learning; neural network; theory of functional connections; partial differential equation
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MDPI and ACS Style

Leake, C.; Mortari, D. Deep Theory of Functional Connections: A New Method for Estimating the Solutions of Partial Differential Equations. Mach. Learn. Knowl. Extr. 2020, 2, 37-55. https://doi.org/10.3390/make2010004

AMA Style

Leake C, Mortari D. Deep Theory of Functional Connections: A New Method for Estimating the Solutions of Partial Differential Equations. Machine Learning and Knowledge Extraction. 2020; 2(1):37-55. https://doi.org/10.3390/make2010004

Chicago/Turabian Style

Leake, Carl, and Daniele Mortari. 2020. "Deep Theory of Functional Connections: A New Method for Estimating the Solutions of Partial Differential Equations" Machine Learning and Knowledge Extraction 2, no. 1: 37-55. https://doi.org/10.3390/make2010004

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