[Paper Review] Distributed physics informed neural network for data-efficient solution to partial differential equations

Vikas Dwivedi, Nishant Parashar+1 more|arXiv (Cornell University)|Jan 1, 2019

Model Reduction and Neural Networks21 references29 citations

TL;DR

This paper proposes a distributed physics-informed neural network (DPINN) that partitions the computational domain into sub-domains, each solved by a shallow local PINN, improving data efficiency and accuracy. DPINN outperforms standard PINNs in solving nonlinear PDEs like Burgers’ and steady-state Navier-Stokes equations, especially in capturing sharp gradients and reducing data requirements.

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.

Motivation & Objective

• Address the limitations of standard PINNs in representing complex functions and handling sharp gradients.
• Overcome the vanishing gradient problem in deep PINNs by using shallow, localized networks.
• Improve data efficiency in solving PDEs by distributing the computational domain and enforcing interfacial continuity.
• Demonstrate the capability of PINN-based methods to solve systems of nonlinear PDEs, such as the Navier-Stokes equations, for the first time directly via physics-informed learning.
• Establish a foundation for future applications in inverse problems and high-dimensional PDEs.

Proposed method

• Partition the computational domain into smaller sub-domains (cells) to enable localized, shallow PINN solutions.
• Use two-layer fully connected neural networks in each cell, reducing depth-related vanishing gradient issues.
• Introduce interfacial flux mismatch penalties in the loss function to enforce continuity across cell boundaries.
• Formulate a composite loss function combining PDE residuals, boundary conditions, initial conditions, and inter-cell continuity constraints.
• Apply automatic differentiation to compute derivatives in the PDE loss terms within each local PINN.
• Train the distributed network using collocation points across the domain, including interfacial points to enforce continuity.

Experimental results

Research questions

• RQ1 Can a distributed PINN architecture improve the representation capability of standard PINNs for complex, high-frequency functions?
• RQ2 Does the DPINN approach reduce data requirements while maintaining or improving solution accuracy for nonlinear PDEs?
• RQ3 Can DPINN successfully solve systems of nonlinear PDEs such as the steady-state Navier-Stokes equations?
• RQ4 How does DPINN perform in capturing sharp gradients and advection of wave packets compared to standard PINN?
• RQ5 To what extent does the interfacial continuity constraint act as a regularizer in improving generalization and data efficiency?

Key findings

• DPINN successfully represents a high-frequency wave packet that standard PINN fails to capture, demonstrating superior representation capability.
• For the Burgers’ equation, DPINN achieves a lower mean squared error (MSE) than PINN with significantly fewer training data points.
• DPINN accurately predicts the steady-state lid-driven cavity flow at Re = 10, with velocity and pressure fields matching well-established benchmark results from Ghia et al. (1982).
• Streamlines, velocity, and pressure contours generated by DPINN show physically consistent flow patterns, confirming the method’s physical plausibility.
• The DPINN approach is the first to directly solve the steady-state Navier-Stokes equations using a physics-informed neural network framework.
• The interfacial continuity constraints in DPINN act as natural regularizers, enhancing generalization and reducing data dependency compared to standard PINN.