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Physics-Informed Neural Networks: Bridging the Divide Between Conservative and Non-Conservative Equations

T0 review · 1 major / 1 minor · reviewed 2026-05-25 · grok-4.3

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Pith's one-line read Physics-informed neural networks perform similarly on shocks whether equations are written in conservative or non-conservative form.

desk verdict This is a straightforward empirical check on whether PINNs care about conservative vs non-conservative form for shocks, but the benchmarks stay inside the class where both forms exist. read the letter →

arxiv 2506.22413 v3 pith:B2M5OOBF submitted 2025-06-27 physics.flu-dyn cs.NAmath.NA

Arun Govind Neelan , Ferdin Sagai Don Bosco , Naveen Sagar Jarugumalli , Suresh Balaji Vedarethinam This is my paper · ORCID
classification physics.flu-dyncs.NAmath.NA
keywords physics-informedneuralnetworksconservativeformnon-conservativeshocksdiscontinuitiesBurgersequationEulerequationscomputationalfluiddynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Traditional discretization methods for fluid flows require equations in conservative form to capture shocks and discontinuities accurately, but many real phenomena involve inherently non-conservative equations. This paper examines whether Physics-Informed Neural Networks share that requirement by testing both formulations on benchmark problems. The investigation covers the Burgers equation along with steady and unsteady Euler equations to assess any sensitivity in how PINNs locate and resolve these features. If PINNs prove insensitive to the choice, they could directly address a wider class of physical problems without forced reformulation into conservative variables.

What carries the argument

Physics-Informed Neural Networks trained on residuals of the PDE in either conservative or non-conservative form, tested for shock-capturing accuracy on Burgers and Euler equations.

What would settle it

A clear and consistent drop in accuracy or increase in shock smearing when switching from conservative to non-conservative form on the Burgers or Euler test cases would falsify the claim of limited sensitivity.

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Extended reading notes

Core claim

The work establishes that PINNs exhibit limited sensitivity to the conservative versus non-conservative formulation of the governing PDEs when applied to flows containing shocks and discontinuities, as demonstrated through systematic comparisons on the Burgers equation and both steady and unsteady Euler equations.

Load-bearing premise

The chosen benchmark problems sufficiently represent the numerical challenges posed by inherently non-conservative PDEs in complex physical phenomena such as multi-phase flows.

Editorial extensions

If this is right

  • PINNs could be applied directly to non-conservative formulations arising in multi-phase or other complex flows without needing to rewrite the equations.
  • The usual restriction that standard numerical solvers impose on equation form would not apply to PINN-based solvers for these problems.
  • Discontinuities in compressible flows could be handled by PINNs regardless of whether the underlying equations are presented in divergence or non-divergence form.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the insensitivity holds, PINNs might reduce the need for specialized conservative-variable transformations that are common in traditional CFD codes.
  • The same tests could be repeated on equations from other domains, such as non-conservative forms in magnetohydrodynamics or viscoelastic flows, to check generality.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The manuscript investigates the sensitivity of Physics-Informed Neural Networks (PINNs) to conservative versus non-conservative PDE formulations for problems involving shocks and discontinuities. It motivates the study by noting that many complex flows (e.g., multi-phase) are inherently non-conservative and conducts empirical tests on the Burgers equation together with steady and unsteady Euler equations.

Significance. If the empirical results were to show that PINNs are insensitive to formulation choice near discontinuities while traditional discretizations are not, the work would offer a practical route to handling non-conservative products without ad-hoc regularization. The current text, however, presents only the intent and benchmark list; no error metrics, loss curves, or comparative tables are supplied, so the significance cannot yet be evaluated.

major comments (1)
  1. [Abstract / Benchmark selection] The central claim that the chosen benchmarks 'provide a comprehensive understanding' of PINN behavior for inherently non-conservative systems rests on an untested assumption. Burgers and both Euler systems remain hyperbolic conservation laws that admit equivalent conservative and non-conservative forms; they therefore do not probe the structural difficulty of non-conservative products that cannot be rewritten in divergence form (the motivating case of multi-phase flows).
minor comments (1)
  1. [Abstract] The abstract states the investigative goal but supplies neither quantitative results nor error metrics; a results-oriented abstract would better convey the paper's contribution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful comments. We address the major comment below and indicate where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract / Benchmark selection] The central claim that the chosen benchmarks 'provide a comprehensive understanding' of PINN behavior for inherently non-conservative systems rests on an untested assumption. Burgers and both Euler systems remain hyperbolic conservation laws that admit equivalent conservative and non-conservative forms; they therefore do not probe the structural difficulty of non-conservative products that cannot be rewritten in divergence form (the motivating case of multi-phase flows).

    Authors: We acknowledge that the referee's observation is correct. The Burgers equation and the Euler equations are hyperbolic conservation laws that admit mathematically equivalent conservative and non-conservative formulations. Our empirical tests therefore examine sensitivity to formulation choice within this class of problems rather than the stricter case of non-conservative products that cannot be expressed in divergence form. We will revise the abstract and introduction to remove the phrasing 'provide a comprehensive understanding' and to explicitly qualify the scope of the benchmarks, noting that they serve as an initial investigation into PINN behavior for shock-containing flows rather than a direct probe of inherently non-rewritable non-conservative systems. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: empirical benchmark study with no derivations or fitted predictions

full rationale

The paper presents an empirical investigation of PINN performance on Burgers and Euler equations in conservative vs. non-conservative forms. No derivation chain, fitted parameters renamed as predictions, self-citation load-bearing premises, or ansatz smuggling is present. The central claim rests on numerical experiments across stated benchmarks rather than any reduction of outputs to inputs by construction. The skeptic concern about benchmark representativeness is a question of external validity, not circularity.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

No free parameters, axioms, or invented entities are described in the abstract; the work is an empirical sensitivity study on existing PINN methodology.

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Cite this review

Pith. "Pith review of Physics-Informed Neural Networks: Bridging the Divide Between Conservative and Non-Conservative Equations." pith.science (2026). https://pith.science/paper/B2M5OOBF

@misc{pith2026250622413,
  author       = {Pith},
  title        = {Pith review of: Physics-Informed Neural Networks: Bridging the Divide Between Conservative and Non-Conservative Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B2M5OOBF}},
  note         = {Machine review of arXiv:2506.22413}
}
read the original abstract

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.

Figures

Figures reproduced from arXiv: 2506.22413 by the authors.

Figure 1
Figure 1. Steps used in Adaptive weight and viscosity PINNs architecture [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Solution of Burgers equation with smooth initial condition [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Solution of Burgers equation with discontinuous initial condition [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Solution of Burgers equation at t = 0.2 s [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Solution of Burgers equation using numerical methods based on non-conservative scheme with artificial [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Pressure solution of Sod shock tube problem over time [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Pressure solution of Sod shock tube problem over time [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: Pressure solution of supersonic flow over wedge [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: Pressure at y=0.5 in supersonic flow over wedge [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

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