The Leray-Burgers equation, used in our study, is a simplified model that retains essential features of the Leray regularization applied to the Navier-Stokes equations. Our research addresses a critical challenge: selecting the appropriate parameter $\alpha$ value, which controls the characteristic wavelength below which smaller-scale physical phenomena are averaged out. While $\alpha$-type regularizations are well-studied, there has been no systematic method for choosing $\alpha$, and the existing rule of thumb ties its selection to numerical schemes and mesh refinements.
Our work aims to decouple $\alpha$ from these numerical factors, allowing for more general and practical application. By employing PINNs, we avoid the constraints of specific meshes and numerical schemes.
Our results demonstrate that the choice of $\alpha$ depends on the initial data, with a practical range of values between 0.01 and 0.05 for continuous initial profiles and between 0.01 and 0.03 for discontinuous profiles.
This study also highlights the effectiveness and efficiency of the Leray-Burgers equation in real practical problems, specifically Traffic State Estimation.