Extreme theory of functional connections: A fast physics-informed neural network method for solving ordinary and partial differential equations

• A new fast, accurate, and robust Physics-Informed Neural Network method has been developed for solving Ordinary Partial Differential Equations For the first time single layer neural networks, trained via Extreme Learning Machine algorithm, Theory of Functional Connections are brought together as a framework The initial boundary conditions always analytically satisfied particular expressions called Constrained Expressions According to input weights bias randomly selected not tuned during training, that reduces training only output computed Least-Squares, linear problems, iterative Least-Squares nonlinear problems We present novel, physics-informed network involving differential equations (DEs), , or X-TFC. proposed is synergy two recently frameworks DEs: TFC, Networks PINN. Here, latent solution DEs approximated by TFC constrained expression employs (NN) free-function. form satisfies constraints DE, while maintaining NN with unconstrained parameters. X-TFC uses single-layer (ELM) algorithm. This choice based on approximating properties ELM algorithm simple least-squares, because trainable parameters weights. methodology was tested over wide range including approximation solutions ordinary (ODEs), systems ODEs, partial (PDEs). results show that, most considered, achieves high accuracy low computational time, even large scale PDEs, without suffering curse dimensionality.