TY - JOUR

T1 - Extreme theory of functional connections

T2 - A fast physics-informed neural network method for solving ordinary and partial differential equations

AU - Schiassi, Enrico

AU - Furfaro, Roberto

AU - Leake, Carl

AU - De Florio, Mario

AU - Johnston, Hunter

AU - Mortari, Daniele

N1 - Funding Information:
This work was partially supported by a NASA Space Technology Research Fellowship, Leake [NSTRF 2019] Grant #: 80NSSC19K1152 and Johnston [NSTRF 2019] Grant #: 80NSSC19K1149.
Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2021/10/7

Y1 - 2021/10/7

N2 - We present a novel, accurate, fast, and robust physics-informed neural network method for solving problems involving differential equations (DEs), called Extreme Theory of Functional Connections, or X-TFC. The proposed method is a synergy of two recently developed frameworks for solving problems involving DEs: the Theory of Functional Connections TFC, and the Physics-Informed Neural Networks PINN. Here, the latent solution of the DEs is approximated by a TFC constrained expression that employs a Neural Network (NN) as the free-function. The TFC approximated solution form always analytically satisfies the constraints of the DE, while maintaining a NN with unconstrained parameters. X-TFC uses a single-layer NN trained via the Extreme Learning Machine (ELM) algorithm. This choice is based on the approximating properties of the ELM algorithm that reduces the training of the network to a simple least-squares, because the only trainable parameters are the output weights. The proposed methodology was tested over a wide range of problems including the approximation of solutions to linear and nonlinear ordinary DEs (ODEs), systems of ODEs, and partial DEs (PDEs). The results show that, for most of the problems considered, X-TFC achieves high accuracy with low computational time, even for large scale PDEs, without suffering the curse of dimensionality.

AB - We present a novel, accurate, fast, and robust physics-informed neural network method for solving problems involving differential equations (DEs), called Extreme Theory of Functional Connections, or X-TFC. The proposed method is a synergy of two recently developed frameworks for solving problems involving DEs: the Theory of Functional Connections TFC, and the Physics-Informed Neural Networks PINN. Here, the latent solution of the DEs is approximated by a TFC constrained expression that employs a Neural Network (NN) as the free-function. The TFC approximated solution form always analytically satisfies the constraints of the DE, while maintaining a NN with unconstrained parameters. X-TFC uses a single-layer NN trained via the Extreme Learning Machine (ELM) algorithm. This choice is based on the approximating properties of the ELM algorithm that reduces the training of the network to a simple least-squares, because the only trainable parameters are the output weights. The proposed methodology was tested over a wide range of problems including the approximation of solutions to linear and nonlinear ordinary DEs (ODEs), systems of ODEs, and partial DEs (PDEs). The results show that, for most of the problems considered, X-TFC achieves high accuracy with low computational time, even for large scale PDEs, without suffering the curse of dimensionality.

KW - Extreme learning machine

KW - Functional interpolation

KW - Least-squares

KW - Numerical methods

KW - Physics-informed neural networks

KW - Universal approximator

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U2 - 10.1016/j.neucom.2021.06.015

DO - 10.1016/j.neucom.2021.06.015

M3 - Article

AN - SCOPUS:85109044892

VL - 457

SP - 334

EP - 356

JO - Neurocomputing

JF - Neurocomputing

SN - 0925-2312

ER -