Neural networks trained on scalar diffusion data predict adaptive coarse basis functions for Schwarz methods, transferring without retraining to linear elasticity and nonlinear p-Laplace problems.
Distributed physics informed neural network for data-efficient solution to partial differential equations
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abstract
The physics informed neural network (PINN) is evolving as a viable method to solve partial differential equations. In the recent past PINNs have been successfully tested and validated to find solutions to both linear and non-linear partial differential equations (PDEs). However, the literature lacks detailed investigation of PINNs in terms of their representation capability. In this work, we first test the original PINN method in terms of its capability to represent a complicated function. Further, to address the shortcomings of the PINN architecture, we propose a novel distributed PINN, named DPINN. We first perform a direct comparison of the proposed DPINN approach against PINN to solve a non-linear PDE (Burgers' equation). We show that DPINN not only yields a more accurate solution to the Burgers' equation, but it is found to be more data-efficient as well. At last, we employ our novel DPINN to two-dimensional steady-state Navier-Stokes equation, which is a system of non-linear PDEs. To the best of the authors' knowledge, this is the first such attempt to directly solve the Navier-Stokes equation using a physics informed neural network.
representative citing papers
The work investigates the sensitivity of PINNs to conservative versus non-conservative PDE formulations when solving benchmark problems that contain shocks and discontinuities.
citing papers explorer
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Learning Adaptive Coarse Spaces Using Transferable Neural Network Models for Linear and Nonlinear Overlapping Domain Decomposition Methods
Neural networks trained on scalar diffusion data predict adaptive coarse basis functions for Schwarz methods, transferring without retraining to linear elasticity and nonlinear p-Laplace problems.
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Physics-Informed Neural Networks: Bridging the Divide Between Conservative and Non-Conservative Equations
The work investigates the sensitivity of PINNs to conservative versus non-conservative PDE formulations when solving benchmark problems that contain shocks and discontinuities.