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.

16 pages, 8 figures

References in corpus (5)

• An Introduction to Variational Autoencoders
• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Adaptive activation functions accelerate convergence in deep and physics-informed neural networks
• Surrogate Modeling for Fluid Flows Based on Physics-Constrained Deep Learning Without Simulation Data
• Inferring solutions of differential equations using noisy multi-fidelity data

Cited by in corpus (22)

• Self-adaptive loss balanced Physics-informed neural networks for the incompressible Navier-Stokes equations
• Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations
• A Physics Informed Neural Network for Time-Dependent Nonlinear and Higher Order Partial Differential Equations
• Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels
• Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology
• Characterizing possible failure modes in physics-informed neural networks
• Multilevel domain decomposition-based architectures for physics-informed neural networks
• Predicting the dynamic process and model parameters of the vector optical solitons in birefringent fibers via the modified PINN
• Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Data
• Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies
• A Method for Computing Inverse Parametric PDE Problems with Random-Weight Neural Networks
• Fourier neural operator for learning solutions to macroscopic traffic flow models: Application to the forward and inverse problems
• Deep Physics Corrector: A physics enhanced deep learning architecture for solving stochastic differential equations
• PICS in Pics: Physics Informed Contour Selection for Rapid Image Segmentation
• Numerical simulation of transient heat conduction with moving heat source using Physics Informed Neural Networks
• Sequential learning based PINNs to overcome temporal domain complexities in unsteady flow past flapping wings
• A composable autoencoder-based iterative algorithm for accelerating numerical simulations
• Eig-PIELM: A Mesh-Free Approach for Efficient Eigen-Analysis with Physics-Informed Extreme Learning Machines
• Multiscale Analysis of Woven Composites Using Hierarchical Physically Recurrent Neural Networks
• Physics-informed time series analysis with Kolmogorov-Arnold Networks under Ehrenfest constraints
• Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs
• Wavefield solutions from machine learned functions