Extremization to fine tune physics informed neural networks for solving boundary value problems

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Abstract

We propose a novel method for fast and accurate training of physics-informed neural networks (PINNs) to find solutions to boundary value problems (BVPs) and initial boundary value problems (IBVPs). By combining the methods of training deep neural networks (DNNs) and Extreme Learning Machines (ELMs), we develop a model which has the expressivity of DNNs with the fine-tuning ability of ELMs. We showcase the superiority of our proposed method by solving several BVPs and IBVPs which include linear and non-linear ordinary differential equations (ODEs), partial differential equations (PDEs) and coupled PDEs. The examples we consider include a stiff coupled ODE system where traditional numerical methods fail, a 3+1D non-linear PDE, Kovasznay flow and Taylor–Green vortex solutions to incompressible Navier–Stokes equations and pure advection solution of 1+1 D compressible Euler equation.
The Theory of Functional Connections (TFC) is used to exactly impose initial and boundary conditions (IBCs) of (I)BVPs on PINNs. We propose a modification to the TFC framework named Reduced TFC and show a significant improvement in the training and inference time of PINNs compared to IBCs imposed using TFC. Furthermore, Reduced TFC is shown to be able to generalize to more complex boundary geometries which is not possible with TFC. We also introduce a method of applying boundary conditions at infinity for BVPs and numerically solve the pure advection in 1+1 D Euler equations using these boundary conditions.
Original languageEnglish
Article number108129
Number of pages29
JournalCommunications in Nonlinear Science and Numerical Simulation
Volume137
Early online date10 Jun 2024
DOIs
Publication statusPublished - Oct 2024

Bibliographical note

This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/).

Keywords

  • Physics informed neural networks
  • Theory of functional connections
  • Boundary value problems
  • PDEs

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