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In this work we present a novel, accurate, and robust physics-informed methodfor solving problems involving parametric differential equations (DEs) calledthe Extreme Theory of Functional Connections, or X-TFC. The proposed method isa synergy of two recently developed frameworks for solving problems involvingparametric DEs, 1) the Theory of Functional Connections, TFC, and thePhysics-Informed Neural Networks, PINN. Although this paper focuses on thesolution of exact problems involving parametric DEs (i.e. problems where themodeling error is negligible) with known parameters, X-TFC can also be used fordata-driven solutions and data-driven discovery of parametric DEs. In theproposed method, the latent solution of the parametric DEs is approximated by aTFC constrained expression that uses a Neural Network (NN) as thefree-function. This approximate solution form always analytically satisfies theconstraints of the DE, while maintaining a NN with unconstrained parameters,like the Deep-TFC method. X-TFC differs from PINN and Deep-TFC; whereas PINNand Deep-TFC use a deep-NN, X-TFC uses a single-layer NN, or more precisely, anExtreme Learning Machine, ELM. This choice is based on the properties of theELM algorithm. In order to numerically validate the method, it was tested overa range of problems including the approximation of solutions to linear andnon-linear ordinary DEs (ODEs), systems of ODEs (SODEs), and partial DEs(PDEs). Furthermore, a few of these problems are of interest in physics andengineering such as the Classic Emden-Fowler equation, the Radiative Transfer(RT) equation, and the Heat-Transfer (HT) equation. The results show that X-TFCachieves high accuracy with low computational time and thus it is comparablewith the other state-of-the-art methods.

Extreme Theory of Functional Connections: A Physics-Informed Neural Network Method for Solving Parametric Differential Equations