Revisiting Conservativeness in Fluid Dynamics: Failure of Non-Conservative PINNs and a Path-Integral Remedy
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914439973830656 |
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| author | Neelan, Arun Govind Bosco, Ferdin Sagai Don Jarugumalli, Naveen Sagar Vedarethinam, Suresh Balaji |
| author_facet | Neelan, Arun Govind Bosco, Ferdin Sagai Don Jarugumalli, Naveen Sagar Vedarethinam, Suresh Balaji |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01968 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Revisiting Conservativeness in Fluid Dynamics: Failure of Non-Conservative PINNs and a Path-Integral Remedy Neelan, Arun Govind Bosco, Ferdin Sagai Don Jarugumalli, Naveen Sagar Vedarethinam, Suresh Balaji Fluid Dynamics Numerical Analysis 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. |
| title | Revisiting Conservativeness in Fluid Dynamics: Failure of Non-Conservative PINNs and a Path-Integral Remedy |
| topic | Fluid Dynamics Numerical Analysis |
| url | https://arxiv.org/abs/2604.01968 |