Temporal Consistency Loss for Physics-Informed Neural Networks

TL;DR

This paper introduces a scaling method for the loss function in physics-informed neural networks, using backward Euler discretization to improve training stability for multiscale PDE problems like Navier-Stokes equations.

Contribution

It proposes a novel loss scaling technique for PINNs that replaces automatic differentiation with backward Euler, enhancing training for complex multiscale PDEs.

Findings

Effective scaling of loss terms improves PINN training stability.

Method performs well on numerical and experimental Navier-Stokes data.

Sensitivity analysis shows robustness to data noise and resolution.

Abstract

Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations in a forward and inverse manner using deep neural networks. However, training these networks can be challenging for multiscale problems. While statistical methods can be employed to scale the regression loss on data, it is generally challenging to scale the loss terms for equations. This paper proposes a method for scaling the mean squared loss terms in the objective function used to train PINNs. Instead of using automatic differentiation to calculate the temporal derivative, we use backward Euler discretization. This provides us with a scaling term for the equations. In this work, we consider the two and three-dimensional Navier-Stokes equations and determine the kinematic viscosity using the spatio-temporal data on the velocity and pressure fields. We first consider numerical datasets to test our method. We test the sensitivity of our method to the time step size, the number of timesteps, noise in the data, and spatial resolution. Finally, we use the velocity field obtained using Particle Image Velocimetry (PIV) experiments to generate a reference pressure field. We then test our framework using the velocity and reference pressure field.