CLINN: Conservation Law Informed Neural Network for Approximating Discontinuous Solutions

Kun Wang; Ruoxi Lu; Tiegang Liu; Weiheng Zeng

REVIEW 4 major objections 3 minor 33 references

CLINN: Conservation Law Informed Neural Network for Approximating Discontinuous Solutions

T0 review · 4 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper claims that a PINN trained with additional implicit-solution, boundedness, and Rankine-Hugoniot losses, plus adaptive refinement driven by an artificial-neuron shock indicator, approximates discontinuous scalar conservation-law so

desk verdict A promising loss-function cocktail for PINNs on conservation laws, but the headline gains likely rely on exact shock-speed data baked into the Rankine-Hugoniot term. read the letter →

arxiv 2509.02091 v2 pith:QGZQP2TT submitted 2025-09-02 math.NA cs.NA

Weiheng Zeng , Kun Wang , Ruoxi Lu , Tiegang Liu This is my paper · ORCID

classification math.NAcs.NA MSC 65M9968T07

keywords scalarconservationlawsphysics-informedneuralnetworksRankine-HugoniotconditionshockdetectionadaptiverefinementBurgersequationBuckley-Leveretttrafficflowmodel

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

CLINN is a recipe for training neural networks to solve scalar conservation laws with moving shocks. It starts from the standard physics-informed loss and adds three terms that encode mathematical facts every entropy solution obeys: the implicit relation u = u0(x - f'(u)t), the bound between the initial data's infimum and supremum, and the Rankine-Hugoniot jump condition across discontinuities. A two-neuron artificial-neuron indicator locates shocks, and an improved residual-based adaptive refinement scheme concentrates training weight near them. On benchmarks from Burgers, traffic flow, and Buckley-Leverett, the paper reports lower mean-squared error than PINN and its immediate variants, with the largest improvement a 99.2% MSE reduction. The practical claim is that conservation-law structure, not just the PDE residual, is what lets PINNs handle discontinuities.

What carries the argument

The load-bearing object is a composite loss L = LPINN + wIM LIM + wBD LBD + wRH LRH. LIM is the implicit-solution residual (predicted u minus u0(x - f'(u)t)); LBD is the boundedness penalty implemented with HardTanh against the initial data's range; LRH is the Rankine-Hugoniot residual at detected discontinuity points, comparing the network's flux jump to the shock speed. The discontinuity set comes from the two-neuron AN indicator, and the improved RAR reweights points according to the governing-equation and implicit-solution residuals outside that set. The combination is what carries the argument: each term injects a different piece of conservation-law structure that the raw PDE residual c

What would settle it

Train CLINN on a Riemann problem whose exact solution is a double contact discontinuity, for example a Buckley-Leverett initial condition arranged so the characteristic speeds on both sides coincide; the paper itself states the AN indicator provably misses such waves, so if the predicted shock location remains accurate anyway, the LRH term is doing less work than claimed, and if it drifts, the central claim fails for this wave type. A second check is to replace the interpolated Phi with the exact discontinuity surface and compare MSE: if the result changes materially, the method's success depe

Watch

Extended reading notes

Core claim

The paper claims that a loss function augmented with three conservation-law-specific terms—the implicit solution form, the boundedness constraint, and the Rankine-Hugoniot jump condition—turns a PINN into a reliable approximator of discontinuous entropy solutions. The implicit solution term penalizes deviations from u = u0(x - f'(u)t) where the solution is smooth; the boundedness term uses HardTanh to discourage values outside the initial range; and the jump-condition term enforces the shock speed computed from the flux jump at points flagged as discontinuities. On top of this, an artificial-neuron indicator marks shock cells and an improved RAR routine increases loss weights at high-residua

Load-bearing premise

The load-bearing premise is that the artificial-neuron indicator tags every real discontinuity, but it provably misses double contact discontinuities, and the discontinuity surface Phi used in the Rankine-Hugoniot loss has no explicit construction from the trained network, so if detection or interpolation is wrong the jump-condition loss is enforced at the wrong places and the reported gains can degrade.

Editorial extensions

If this is right

For scalar conservation laws with convex, concave, and nonconvex fluxes, CLINN reproduces shock locations and solution profiles with lower MSE than PINN, IFNN, and PINN-WE across the reported benchmarks.
The boundedness loss suppresses spurious oscillations on the downstream side of shocks, visible in the LWR and nonconvex cases.
Combining the implicit-solution and Rankine-Hugoniot terms selects the physically admissible solution in regions where the implicit form is multivalued, such as the rarefaction-shock interaction in the LWR test.
The improved RAR reweights high-residual points near discontinuities, adding further error reduction after the first training phase; the ablation model without RAR performs worse after 5000 epochs.
The framework extends to 2D by applying the AN indicator via dimensional splitting, demonstrated on the 2D Burgers equation.

Reading between the lines

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

If the AN indicator's blind spot for double contact discontinuities is patched or replaced by a detector that includes characteristic-speed coincidence, the same loss formula should cover all wave types, since the Rankine-Hugoniot term itself does not require convexity.
Because the paper shows CLINN trails IFNN on the simple periodic Burgers case, an adaptive switch that activates LRH only when a shock is actually detected could avoid the optimization slowdown while preserving accuracy on harder cases.
The scalar-law construction suggests a route to systems: enforce the vector Rankine-Hugoniot condition componentwise along each wave family, though the paper leaves systems to future work.
A testable extension is to replace the unstated interpolation of the discontinuity surface Phi with an exact or independently computed surface and measure how much of the MSE gain comes from that construction rather than from the loss terms themselves.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Here's my read of CLINN. The combination is genuinely new: implicit-solution loss plus boundedness plus Rankine-Hugoniot jump term, with an AN-based shock indicator driving an RAR reweighting scheme. That packaging, tested across Burgers, LWR, and Buckley-Leverett with convex, concave, and non-convex fluxes, is a solid piece of engineering work. The authors are also honest where it counts: they admit the AN indicator misses double contact discontinuities, and they report that CLINN is slightly worse than IFNN on case (1A). The method description is detailed enough to reimplement, and the ablation "CLINN w/o RAR" helps isolate the adaptive weighting contribution.

But there is a load-bearing problem. The Rankine-Hugoniot term needs the shock speed s(x_j,t_j). The paper only says that the discontinuity surface Φ "can be obtained through interpolation" and never specifies how s is computed from the network or from the detected points. In every numerical experiment, the exact Φ is supplied explicitly (Eq. 4.6, 4.26, etc.). If the implementation uses that exact Φ to evaluate s, then LRH is a supervised term using the exact shock location and speed. The reported 87%–99% improvement over PINN would then be the result of privileged information, not a self-contained conservation-law constraint. The acknowledged double-contact failure is consistent with this: for case (3A), the AN indicator provably cannot identify the contact, so the only way LRH could be active is through the exact Φ. This doesn't make the paper useless, but it fundamentally changes what the headline numbers mean.

Other problems are more routine. There are no repeated runs or error bars; the best checkpoint is selected by test MSE against the exact solution; and the phrase "exact conservation" overstates what a soft loss penalty provides. The comparison omits recent weak-form and thermodynamically consistent PINN baselines, though the three baselines they do use are appropriate.

My overall verdict: this is a serious paper with a real idea but a central methodological gap. The right fix is either to rerun without exact shock data (estimating s from the network or from a separate detection step) or to reframe CLINN as using exact shock information as an additional supervised input. As written, the forward-solver claim is not supported.

I'd send it to peer review rather than desk-reject: the combination of ideas merits careful referee scrutiny, and the flaw is fixable. I'd also want code and error bars before publishing. If I were working in this area, I'd cite it as an example of how easy it is to leak exact solution information into a supposedly physics-informed loss.

Best, [Your name]

Referee Report

4 major / 3 minor

Summary. The paper proposes CLINN, a PINN variant for scalar conservation laws that augments the usual governing-equation and initial/boundary losses with three conservation-law-informed terms: an implicit-solution loss LIM, a boundedness loss LBD, and a Rankine-Hugoniot jump-condition loss LRH. It also uses the Feng–Liu artificial-neuron indicator to detect discontinuities and a residual-based adaptive refinement (RAR) scheme to reweight collocation points near shocks. The method is tested on 1D inviscid Burgers, LWR traffic flow, Buckley-Leverett, and a 2D Burgers problem, reporting large MSE reductions relative to PINN (up to 99.2%) and generally better accuracy than IFNN and PINN-WE.

Significance. If the method were self-contained, the reported gains would be practically valuable for approximating discontinuous solutions of scalar conservation laws. The paper gives a detailed, reimplementable experimental setup, compares against three baselines, includes an RAR ablation, and honestly acknowledges the AN indicator's limitation for double contact discontinuities. However, the central novelty—the Rankine-Hugoniot loss—appears to rely on externally supplied exact shock-speed information in every benchmark, which would make the comparison with PINN unfair and undermine the claim of a practical forward solver. The evaluation also uses test-set oracle checkpoint selection with no error bars. These issues are load-bearing for the paper's central claim.

major comments (4)

[§3.1, Eq. (3.1)–(3.2)] The Rankine-Hugoniot term LRH is evaluated at detected discontinuity points and penalizes |(f(û_L)-f(û_R))/(û_L-û_R) - s(xj,tj)|. The manuscript never specifies how s(xj,tj) is obtained from the trained network or from the AN indicator. In all numerical experiments the exact discontinuity surface is available and stated: Eq. (4.6) for (1A), Eq. (4.26) for the 2D case, and exact Riemann solutions for the other cases. If s is taken from these analytic expressions, LRH is a supervised term using analytic shock data. If, alternatively, s is computed from the same network values û_L and û_R, the term is identically zero and imposes no constraint. Thus the central loss term either requires privileged information or is vacuous. A practical forward PINN variant needs an autonomous way to obtain s; the current text does not provide one.
[§3.2, test case (3A)] The AN indicator is acknowledged to fail on double contact discontinuities because λL−λR vanishes identically. The Buckley-Leverett case (3A) has exactly such a double contact, since λ(1)=λ(0)=0 for the flux in Eq. (4.15). Yet CLINN is reported to outperform on (3A) (Table 4), and Fig. 9(h) is described as agreeing with exact shock positions. This is internally inconsistent unless the discontinuity set PD is taken from the exact solution rather than from the indicator. Please clarify how PD was constructed for this case and, more generally, what 'Φ(x,t) can be obtained through interpolation' means in §3.1.
[§4.1, 'Comparisons'] The evaluation selects checkpoints using the exact solution: 'We save each model that achieves the smallest MSE between the predicted and the exact solutions after 1000 epochs.' Because the same test MSE is the evaluation metric, the reported MSEs in Tables 2–5 and the improvement ratios of Eq. (4.1) are optimistically biased. No multi-seed statistics or error bars are provided. The comparison would be much more convincing with fixed-epoch models, early stopping on a validation set, or repeated independent runs.
[Abstract and §5] The phrase 'enforcing exact conservation properties' is an overclaim. The boundedness term LBD and the Rankine-Hugoniot term LRH are soft penalties in the loss; they do not enforce exact conservation either locally or globally. The paper provides no conservation-error measurement, such as the evolution of the spatial integral of the solution. Please revise the wording and, ideally, report a conservation-error metric to support the claim.

minor comments (3)

[Eq. (2.12)] There is a typo in the loss definition: 'wICLBC' should presumably be 'wBCLBC'. Please correct.
[Abstract / §4.3 / tables] Minor language issues: 'numeral oscillations' should be 'numerical oscillations'; 'Two-Dimenstional' should be 'Two-Dimensional'; 'Comparation' in table captions should be 'Comparison'.
[§3.1, LBD definition] The notation HardTanh²(û; inf u0, sup u0) is unclear. If it means the squared distance to the interval [inf u0, sup u0], please define it explicitly, because the current notation could be read as the square of the HardTanh output.

Circularity Check

1 steps flagged · score 6.0 of 10

CLINN's shock-location accuracy is partially circular: the Rankine-Hugoniot loss term is defined using the exact shock speed/surface from the analytic benchmarks, so the reported discontinuity predictions are supervised by the target data.

self definitional [Section 3.1, Eq. (3.1)-(3.2); Section 4.2.1, Eq. (4.6); Section 4.3, Eq. (4.26)]
"LRH,j = |(f(ûj,L) − f(ûj,R))/(ûj,L − ûj,R) − s(xj, tj)|; 'the discontinuity surface Φ(x, t) can be obtained through interpolation; s(x, t) is defined in Proposition 2.1'; for (1A): 'The trajectory of the discontinuity is given by (4.6) Φ(x, t) = x − t/2 − (2k + 1) = 0.'"

The LRH term is minimized only when the network's Rankine-Hugoniot ratio equals s(xj,tj), the normal velocity of the exact discontinuity surface. The paper gives no procedure for computing s from the trained network or from the AN indicator; it says only that Φ 'can be obtained through interpolation.' In every benchmark, the exact Φ or exact Riemann solution is supplied explicitly (e.g., Eq. (4.6) for 1A and Eq. (4.26) for the 2D case). Thus the discontinuity locations that CLINN is credited with predicting are, by construction, inputs to the loss. The large reported MSE reductions over PINN therefore partly reflect supervision by analytic shock data rather than a self-contained forward solve. The admitted double-contact failure is consistent: where no exact Φ is available, LRH cannot corr

full rationale

The implicit-solution term LIM and boundedness term LBD are self-contained penalty constraints derived from the conservation-law properties (2.2)-(2.3); using them in a loss and then measuring external MSE is not circular. The AN indicator is imported from Feng et al., which includes two of the present authors, but it is published prior work and is used as a tool rather than as a uniqueness argument. The significant circularity is the Rankine-Hugoniot term: Eq. (3.1)/(3.2) requires s(x,t), the exact shock speed, yet no method is described for computing s from the network's predictions, and each experiment supplies exact Φ/s. Consequently, the central claim of superior shock-location accuracy is at least partially forced by the target data. Because the evaluation is external and the other loss terms retain independent content, the paper is partially circular rather than a tautology; a fully self-contained version would need to estimate Φ and s from the network and then demonstrate accuracy without analytic shock data.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The method introduces no new physics or entities; it rests on standard conservation-law facts and on hand-chosen hyperparameters that carry much of the reported gain.

free parameters (4)

Loss weights wIM, wBD, wRH, wIC, wBC = 10000, 10000, 100, 1000, 10
Set by hand in Section 4.1; no sensitivity study; central to the reported MSE gains.
RAR weights wEQ, wIF and Npt = 33, 16, 500
Chosen in Section 4.1; the refinement schedule modifies training dynamics near discontinuities.
Network architecture (5x100, lr=1e-4) = 5 layers, 100 neurons, Adam lr=1e-4
Standard but arbitrary choices; no ablation for architecture.
Shock-probe offset h in LRH = unspecified
Eq. (3.2) defines h as 'a small increment' without a numerical value; affects jump-condition computation.

assumptions (4)

standard math Entropy solution is the physically relevant weak solution (Oleinik condition, Eq. 2.6)
Invoked in Section 2.1 to justify uniqueness and the use of Rankine-Hugoniot condition.
standard math All solutions stay within [inf u0, sup u0] (Eq. 2.3)
Standard maximum principle for scalar conservation laws cited to [4], used to define LBD.
standard math Implicit solution u = u0(x - lambda(u)t) holds in smooth regions (Eq. 2.2)
Characteristic theory result; used as LIM loss in Eq. (3.1).
domain assumption AN-based shock indicator parameters W=10, M=-12, C=-1 are transferable to new problems
The indicator parameters come from Feng et al. training and are used without retraining; if the network's predicted u at cell centers does not provide lambda accurately, detection fails.

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Cite this review

Pith. "Pith review of CLINN: Conservation Law Informed Neural Network for Approximating Discontinuous Solutions." pith.science (2026). https://pith.science/paper/QGZQP2TT

@misc{pith2026250902091,
  author       = {Pith},
  title        = {Pith review of: CLINN: Conservation Law Informed Neural Network for Approximating Discontinuous Solutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QGZQP2TT}},
  note         = {Machine review of arXiv:2509.02091}
}

read the original abstract

Physics-informed Neural Network (PINN) faces significant challenges when approximating solutions to conservation laws, particularly in ensuring conservation and accurately resolving discontinuities. To address these limitations, we propose Conservation Law-informed Neural Network (CLINN), a novel framework that incorporates the boundedness constraint, implicit solution form, and Rankine-Hugoniot condition of scalar conservation laws into the loss function, thereby enforcing exact conservation properties. Furthermore, we integrate a residual-based adaptive refinement (RAR) strategy to dynamically prioritize training near discontinuities, substantially improving the network's ability to capture sharp gradients. Numerical experiments are conducted on benchmark problems, including the inviscid Burgers equation, the Lighthill-Whitham-Richards (LWR) traffic flow model, and the Buckley-Leverett problem. Results demonstrate that CLINN achieves superior accuracy in resolving solution profiles and discontinuity locations while reducing numeral oscillations. Compared to conventional PINN, CLINN yields a maximum reduction of 99.2% in mean squared error (MSE).

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Study

J. Abbasi, A. D. Jagtap, B. Moseley, A. Hiorth, and P. Ø. Andersen , Challenges and advancements in modeling shock fronts with physics-informed neural networks: A review and benchmarking study , arXiv preprint arXiv:2503.17379, (2025)

work page Pith review arXiv 2025

[2] [2]

S. E. Buckley and M. Leverett , Mechanism of fluid displacement in sands , Transactions of the AIME, 146 (1942), pp. 107–116

work page 1942

[3] [3]

J. M. Burgers, Mathematical examples illustrating relations occurring in the theory of turbu- lent fluid motion , in Selected Papers of JM Burgers, Springer, 1995, pp. 281–334

work page 1995

[4] [4]

Chang and L

T. Chang and L. Hsiao , The Riemann problem and interaction of waves in gas dynamics , NASA STI/Recon Technical Report A, 90 (1989), p. 44044

work page 1989

[5] [5]

J. Chen, X. Chi, Z. Yang, et al., Bridging traditional and machine learning-based algorithms for solving PDEs: the random feature method , J Mach Learn, 1 (2022), pp. 268–298

work page 2022

[6] [6]

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arXiv 2023

[7] [7]

Cockburn and C.-W

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work page 1989

[8] [8]

Cockburn and C.-W

B. Cockburn and C.-W. Shu , The Runge-Kutta local projection-discontinuous-Galerkin fi- nite element method for scalar conservation laws , ESAIM: Mathematical Modelling and Numerical Analysis, 25 (1991), pp. 337–361

work page 1991