Temporal consistency loss for physics-informed neural networks

Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations (PDEs) in a forward and inverse manner using neural networks. However, balancing individual loss terms can be challenging, mainly when training these networks for stiff PDEs and scenarios requiring enforcement of numerous constraints. Even though statistical methods can be applied to assign relative weights to the regression loss for data, assigning relative weights to equation-based loss terms remains a formidable task. This paper proposes a method for assigning relative weights to the mean squared loss terms in the objective function used to train PINNs. Due to the presence of temporal gradients in the governing equation, the physics-informed loss can be recast using numerical integration through backward Euler discretization. The physics-uninformed and physics-informed networks should yield identical predictions when assessed at corresponding spatiotemporal positions. We refer to this consistency as “temporal consistency.” This approach introduces a unique method for training physics-informed neural networks (PINNs), redefining the loss function to allow for assigning relative weights with statistical properties of the observed data. In this work, we consider the two- and three-dimensional Navier–Stokes equations and determine the kinematic viscosity using the spatiotemporal data on the velocity and pressure fields. We consider numerical datasets to test our method. We test the sensitivity of our method to the timestep size, the number of timesteps, noise in the data, and spatial resolution. Finally, we use the velocity field obtained using particle image velocimetry experiments to generate a reference pressure field and test our framework using the velocity and pressure fields.