Naveen Sagar Jarugumalli

Arun Govind Neelan, Ferdin Sagai Don Bosco, Naveen Sagar Jarugumalli et al.

In the realm of computational fluid dynamics, traditional numerical methods, which heavily rely on discretization, typically necessitate the formulation of partial differential equations (PDEs) in conservative form to accurately capture shocks and other discontinuities in compressible flows. Conversely, utilizing non-conservative forms often introduces significant errors near these discontinuities or results in smeared shocks. This dependency poses a considerable limitation, particularly as many PDEs encountered in complex physical phenomena, such as multi-phase flows, are inherently non-conservative. This inherent non-conservativity restricts the direct applicability of standard numerical solvers designed for conservative forms. This work aims to thoroughly investigate the sensitivity of Physics-Informed Neural Networks (PINNs) to the choice of PDE formulation (conservative vs. non-conservative) when solving problems involving shocks and discontinuities. We have conducted this investigation across a range of benchmark problems, specifically the Burgers equation and both steady and unsteady Euler equations, to provide a comprehensive understanding of PINNs capabilities in this critical area.

Arun Govind Neelan, Ferdin Sagai Don Bosco, Naveen Sagar Jarugumalli et al.

The choice between conservative and non-conservative formulations is a fundamental dilemma in CFD. While non-conservative forms offer intuitive modeling in primitive variables, they typically produce erroneous shock speeds. This paper critically analyzes these formulations, contrasting classical failures against the capabilities of Physics-Informed Neural Networks (PINNs). Using the Adaptive Weight and Viscosity (PINNs-AWV) architecture, we evaluate cases ranging from shallow water equations to unsteady 1D and 2D Euler equations. Results reveal a significant dichotomy: while PINNs-AWV restores physical fidelity in scalar and steady systems, standard non-conservative PINNs fail in unsteady systems like the Sod shock tube. We demonstrate this failure stems from non-vanishing source terms introduced by viscous regularization, which violate the Rankine--Hugoniot jump conditions. To resolve this, we implement a path-integral framework based on Dal Maso--LeFloch--Murat (DLM) theory. By incorporating path-consistent losses in PINNs (PI-PINN) and using path-conservative numerical schemes, we successfully recover correct shock speeds within non-conservative frameworks. Our results prove the path-integral approach provides a rigorous mathematical bridge for physical accuracy in both classical and machine learning solvers, enabling primitive-variable formulations in transient, high-speed simulations.