Distributed physics informed neural network for data-efficient solution to partial differential equations

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.

Figures (8)

• Figure 1: Schematic of a minimal deep neural network
• Figure 2: DPINN architecture for the full domain and an individual cell.
• Figure 3: Schematic of DPINN architecture
• Figure 4: Solution of the advection equation \ref{['eq:advection']} obtained using PINN at three time instants: (a) t=0, (b) t=0.1 and (c) t=0.2. (solid line represents PINN solution and dashed lines represents the exact solution)
• Figure 5: Solution of the advection equation \ref{['eq:advection']} obtained using DPINN at three time instants: (a) t=0, (b) t=0.1 and (c) t=0.2. (solid line represents DPINN solution and dashed lines represents exact solution)