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Graph neural operator learns shock waves in traffic flow with one forward pass.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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T0 review · deepseek-v4-flash

2026-08-02 07:26 UTC pith:QRVPTVYO

load-bearing objection Worth engaging for the receptive-field necessary condition and the physics-gated graph architecture, but the headline accuracy claims rest on a single-resolution, in-distribution comparison that does not yet support 'best reported accuracy.' the 3 major comments →

arxiv 2607.20541 v1 pith:QRVPTVYO submitted 2026-07-10 cs.LG cs.AI

HypNO: A Graph-Based Neural Operator with Physics-Informed Message Passing for Hyperbolic Conservation Laws

Dimitrije \v{Z}drale , Cassie An Jeng , Katie Wang , Sonia Vanier , Alexandre Bayen , Hossein Nick Zinat Matin This is my paper
classification cs.LG cs.AI MSC 35L6565M0868T07
keywords hyperbolic conservation lawsneural operatorgraph neural networkmessage passingphysics-informed gatingLWR traffic flowARZ traffic flowshock capturing
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proposes HypNO, a graph neural operator that maps an initial condition for a hyperbolic conservation law directly to the full space-time solution in a single forward pass, without sequential time stepping. The central claim is that by structuring message passing on a causal space-time cell graph with finite-volume-style edge features (flux, characteristic speed, Rankine-Hugoniot speed, upwind flags) and gating messages with upwind, entropy, and CFL-style gates, the model can keep shocks and contacts sharp instead of smearing them. On the LWR and ARZ traffic-flow benchmarks, the paper reports the lowest mean absolute error against exact Lax-Hopf and wave-front-tracking entropy solutions in every stratified cell, with a three- to fourfold error reduction over FNO and classical schemes on ARZ and a tighter per-sample error spread. An appendix claims a necessary design rule — the receptive field must cover the domain of dependence, L*kx*dx >= c_max*T — below which predicted shock tracks freeze because the determining initial data leave the receptive field. If correct, the work offers a mechanistic guideline for building shock-aware neural operators and a fast surrogate for real-time traffic-flow prediction.

Core claim

The central discovery is that a hyperbolicity-aware graph architecture — causal space-time connectivity, physics-derived edge features, and multiplicative gates encoding upwind direction, entropy admissibility, and CFL-limited temporal reach — suffices to learn an operator that keeps shocks and contact discontinuities sharp in a single forward pass. On stratified LWR and ARZ evaluations with exact entropy ground truth, the model attains the lowest density MAE in every (initial-condition family, segment-count) cell, including out-of-distribution segment counts; on ARZ the pooled density error is three to four times smaller than FNO, Godunov, HLL, and WENO5, with a markedly tighter per-sample

What carries the argument

The carrying object is a space-time graph whose nodes are finite-volume cells (x_i, t_n) and whose edges form a causal stencil N(kx,kt): nodes only receive messages from the same or earlier time levels, and the initial condition is broadcast to all time levels before encoding. Each directed edge carries finite-volume interface quantities — flux, characteristic speed, Rankine-Hugoniot speed, upwind indicators, entropy-violation indicators, and a CFL ratio — and each message is weighted by a product gate g = g_up * g_ent * g_time, with separate message MLPs for adjacent (finite-volume-like) and non-adjacent edges. A shared decoder produces physical probes at every layer for deep supervision. T

Load-bearing premise

The benchmark scores every method on the same single 128-cell grid with exact in-family targets, while classical schemes carry fixed O(dx) shock-diffusion error at that resolution, so the reported 'best accuracy' compares a sharp-target-trained network against diffusive classical schemes on the training family rather than demonstrating general solver superiority.

What would settle it

Run a matched-cost or resolution-converged evaluation on smooth, off-family initial data (e.g., random smooth profiles or sine staircases not drawn from the piecewise-constant family) and at successively refined grids; if the three- to fourfold error margin shrinks or classical schemes match it at their native resolution, the 'best reported accuracy' claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Shock-aware neural operators are feasible: a carefully biased graph architecture can represent discontinuous solutions as sharp fronts in a single pass, countering the blurring tendency of global spectral operators.
  • The receptive-field condition provides a concrete formula for sizing message-passing depth and stencil width for any hyperbolic PDE, needing only the maximum characteristic speed and final time.
  • The upwind, entropy, and CFL gates show how to inject discrete finite-volume intuition into a learned operator without hard-coding a particular numerical scheme.
  • Zero-shot performance on unseen segment counts suggests the operator generalizes to more complex piecewise-constant initial conditions than those seen in training, within the same data family.
  • The consistently tight per-sample error spread indicates uniform accuracy across initial conditions, which matters for deploying surrogates in safety-critical or real-time settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the reported margin is measured on a single 128-cell grid while the network is trained on exact solutions from the same piecewise-constant/Riemann family as the tests; classical schemes carry O(dx) shock-diffusion error that a sharp-target-trained network never incurs, so a matched-cost or resolution-converged comparison on off-family smooth data could shrink the apparent adva
  • Editorial extension: the receptive-field rule is a general design principle for any local message-passing neural operator on space-time graphs, potentially applicable to MeshGraphNets or other learned PDE solvers that need to respect domains of dependence.
  • Editorial extension: the entropy gate's use of the Lax condition and the ARZ router between 1-wave and 2-contact jumps could be adapted to other hyperbolic systems (shallow water, Euler) where multiple wave families and contact discontinuities matter.
  • Editorial extension: the sharpness advantage suggests a testable hypothesis that training on multi-resolution exact data (or using the CFL gate to adapt stencils to local wave speeds) would let the model approach the accuracy of adaptive or high-resolution numerical methods.

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

3 major / 6 minor

Summary. The paper proposes HypNO, a graph-based neural operator for scalar hyperbolic conservation laws. It maps an initial condition to the full space-time solution by stacking the initial data on a causal space-time graph and applying message passing whose edge features and scalar gates are built from finite-volume quantities: flux, characteristic speeds, Rankine-Hugoniot speeds, upwind flags, entropy-admissibility indicators, and CFL-type temporal weights. The architecture is evaluated on the LWR scalar traffic model and the ARZ system against FNO, WENO5, Godunov, and HLL, using exact Lax-Hopf and wave-front-tracking ground truth, stratified by initial-condition family and segment count. The paper reports that HypNO attains the lowest density MAE in every evaluated cell, with three- to fourfold reductions over baselines on ARZ, and derives a necessary receptive-field condition L k_x dx >= c_max T (Eq. 49) for shock tracking.

Significance. If the empirical claims are established, HypNO would be a useful contribution: the physics-gated message-passing design is clearly motivated, the receptive-field condition is a parameter-free design rule, and benchmarking against exact entropy solutions with per-cell standard deviations over 20 samples is a real strength. The architecture and the necessary-condition analysis are worth publishing. However, the central 'best reported accuracy' claim currently rests on a single-resolution, in-distribution comparison protocol, and the freeze-sweep evidence for Eq. (49) is asserted without data. The accuracy claim is therefore not yet established, although the underlying architecture and design rule remain plausible.

major comments (3)
  1. [§5.1–5.2, Tables 1–5] The headline comparison is performed at a single resolution (nx=128, dx=1/64) against exact entropy solutions. Classical finite-volume schemes carry O(dx) shock-diffusion error at this grid, and FNO smooths fronts; HypNO is trained on sharp exact solutions drawn from the same stratified piecewise-constant/Riemann families used at test time (Appendix C). The reported 3–4x margins therefore likely measure in-family front sharpness rather than general solver superiority. To support the 'best reported accuracy' framing, the authors should add matched-resolution comparisons (e.g., Godunov/HLL/WENO5 at dx=1/256 or 1/512, downsampled to the evaluation grid) and/or matched-cost comparisons (wall-clock, FLOPs, or number of evaluations), and report performance on an off-family initial-condition set.
  2. [§5.1–5.2, Tables 1, 3–4] The out-of-distribution claim is overstated. For piecewise-constant LWR and ARZ, OOD only changes the number of segments {8,20,25,30} while the samples are drawn from the same stratified sampler and initial-condition families as training; the Riemann families are entirely in-distribution. This tests extrapolation in segment count, not zero-shot generalization to new initial-condition families or to non-piecewise-constant data. The text should either soften the OOD/generalization wording or add held-out IC families (such as the sine-staircase family already used in ARZ training, or smooth initial data for LWR) and cross-family transfer experiments.
  3. [Appendix B.4, Eq. (49)] The necessary-condition derivation leading to Eq. (49) is sound and is a useful design principle. However, the claim that the bound is 'sharp in practice' is supported only by an asserted sweep: 'Sweeps over Riemann initial data confirmed the predicted linear scaling of t_freeze with 1/|s|, with slope L kx dx.' No figure, table, or numerical values are provided. Since this diagnostic is used to justify the selected (L, kx) configurations, the authors should include the freeze measurements (t_freeze versus 1/|s|, with the predicted slope) or clearly label this part as a hypothesis rather than a confirmed finding.
minor comments (6)
  1. [Table 1] In the Riemann block, every row is labeled '2' although the rows appear to correspond to different stratified bins or samples. Use distinct bin labels or collapse the rows; as printed, the table is confusing.
  2. [Appendix C.2.3] There is a duplicated and malformed phrase around '4 |{z} contact strength = 32 cells, where...' that repeats several lines verbatim. The text should be cleaned and the correct cell-count equation restored.
  3. [Eq. (14), §B.2.1] The text says the edge feature vector has 8 slots, but Equation (14) lists fewer entries. Either include the sign(s) term mentioned in Remark 1 in the equation or correct the slot count.
  4. [§6, §2.2] The paper claims resolution generalization ('generalizes across resolutions') but reports no experiments at different nx. Please either add coarser/finer-grid evaluations or remove the claim from the introduction/conclusion.
  5. [Table 7] Network configuration is given for the ARZ models but not for the LWR model. Provide the equivalent hyperparameter table for LWR, including learning rate, weight decay, and the exact values of tau, gamma, kappa, and delta.
  6. [General] No code, data seeds, or trained-model artifacts are provided. Exact reproducibility of the tables would be substantially improved by releasing the generation scripts and model configuration.

Circularity Check

0 steps flagged

No significant circularity: the central empirical claims are benchmarked against external exact references and the receptive-field rule is derived from hyperbolicity and the stencil definition, not fitted.

full rationale

The paper's main derivation chain is the receptive-field necessary condition in Section B.4 (Eq. 49), which follows from the definition of the network's receptive field after L layers (Eq. 48) and the hyperbolic domain of dependence ∠{x : |x − x_i| ≤ c_max T}. This is a parameter-free necessary condition, not a fitted quantity; the freeze-sweep observation is an empirical illustration, not the source of the bound. The benchmark comparisons in Section 5 are all scored against external exact references (Lax–Hopf for LWR, wave-front tracking / exact Riemann solver for ARZ) and against independent baselines (FNO, WENO5, Godunov, HLL). The model is trained on exact solutions from the same stratified families as the test sets, but the test samples are held out and the metric is against ground truth; this is standard in-distribution generalization, not a prediction forced by construction. The physics gates are computed from the model's own decoded probes, which is self-referential as an architecture, but the probes are supervised intermediate quantities and the final output is assessed against exact solutions; no equation of the paper equals its input by construction. The self-citations ([2], [19]) appear only as related work describing prior neural finite-volume schemes; they are not load-bearing for the paper's central claims and no uniqueness theorem is imported from them. Evaluation-protocol concerns about single-resolution or matched-cost comparisons are correctness/robustness issues, not circularity. No circular step can be exhibited from the paper's equations or citations.

Axiom & Free-Parameter Ledger

5 free parameters · 6 axioms · 0 invented entities

The paper introduces no new physical entity, force, or conserved quantity; the upwind/entropy/CFL gates and the cell graph are network mechanisms, not entities with independent falsifiable handles. The only new mathematical object is the necessary-condition inequality (Eq. 49), which is a design rule rather than an entity. Free parameters are limited to three learned gate scalars per PDE (tau, gamma, kappa), the hand-chosen shock-mask thresholds (Section F), and the unspecified RH-fallback threshold delta (Eq. 40). Data-sampler constants (Delta_min, Delta_max, number of bins) shape the training distribution but are not fit to the result.

free parameters (5)
  • Upwind-gate temperature tau (per PDE; per family in ARZ) = learned: tau = softplus(theta_tau) + eps
    B.2.3 / B.3.3. Controls the sharpness of the sigmoid upwind gate; a learned scalar, not hand-set.
  • Entropy-gate floor gamma = learned: gamma = sigmoid(theta_gamma)
    B.2.4 / B.3.4. Floor weight applied to entropy-violating interfaces; learned.
  • CFL-gate decay kappa = learned: kappa = softplus(theta_kappa)
    B.2.5 / B.3.5. Exponent governing suppression of super-CFL edges; learned.
  • Shock-band detection threshold tau = 0.06, TV multiplier alpha = 1.5, dilation b = 2 = tau = 0.06; alpha = 1.5; b = 2
    Section F. Hand-chosen; shapes the shock-band diagnostic rather than the headline full-domain MAE. Masks are reused across methods, so the comparison is fair, but the threshold choice affects the reported shock-band margins.
  • ARZ Rankine-Hugoniot fallback threshold delta = unspecified
    Eq. (40). Density-jump threshold below which the secant RH speed is replaced by lambda_1; the value is not given in the paper.
axioms (6)
  • domain assumption The Lax entropy condition selects the physically relevant weak solution among non-unique weak solutions
    Used throughout Section 3 and B.2.4-B.3.4 to justify the entropy gates and the choice of ground truth.
  • domain assumption The Lax-Hopf formula gives the exact entropy solution for LWR with Greenshields flux f(rho) = rho(1-rho)
    Section 5.1, Figure 1. Ground truth for LWR training and evaluation.
  • domain assumption Wave-front tracking with rarefaction fans of width delta = 0.1*dx approximates the ARZ entropy solution to machine precision
    Section C.2.2. Ground truth for the multi-segment ARZ family; agreement with the exact Riemann solver claimed at O(10^-7) on Riemann data.
  • standard math Domain of dependence of the entropy solution at (x,t) is the cone |x'-x| <= c_max*t, with c_max = max|lambda| (1 for LWR, <=1 for ARZ)
    Appendix B.4, Eq. (49). The load-bearing premise of the receptive-field necessary condition.
  • domain assumption ARZ pressure p(rho) = rho (gamma = 1) with v in [0,1] and rho in (0.1, 0.9)
    Sections 3.1.2 and C.2. The chosen pressure law and state ranges set the characteristic speeds and the c_max bound; restricts generality to this traffic-flow regime.
  • domain assumption Supervised training on exact solutions (with deep supervision) induces a generalizing operator rather than memorization
    Sections 4.7 and 5. The OOD segment-count results are the main evidence; no explicit train-vs-test gap or memorization analysis is shown.

pith-pipeline@v1.3.0-alltime-deepseek · 36138 in / 25006 out tokens · 275431 ms · 2026-08-02T07:26:54.086075+00:00 · methodology

0 comments
read the original abstract

We introduce HypNO, a graph-based neural operator for scalar hyperbolic conservation laws. HypNO operates directly on a space-time graph of finite-volume cells and uses adjacency-factored, physics-informed message passing to respect upwinding and entropy admissibility near shocks. We benchmark the architecture on the Lighthill-Whitham-Richards (LWR) and Aw-Rascle-Zhang (ARZ) traffic-flow models, a stress test for operator-learning methods because of their simultaneous global transport and shock formation. HypNO predicts solution snapshots accurately across a range of initial conditions while capturing the shocks and discontinuities of the solution.

Figures

Figures reproduced from arXiv: 2607.20541 by Alexandre Bayen, Cassie An Jeng, Dimitrije \v{Z}drale, Hossein Nick Zinat Matin, Katie Wang, Sonia Vanier.

Figure 1
Figure 1. Figure 1: Exact LWR solutions for piecewise-constant initial data of increasing complexity. Each column corre [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Exact ARZ solutions for piecewise-constant initial data of increasing complexity, with pressure [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: HypNO. (a) The model lifts u0(x) onto a space-time grid and predicts the full field u over the (t, x)- coordinates through L physics-gated message-passing layers. (b) One layer: messages on a stencil are weighted by a physics gate g = gupgentgtime. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Quantitative and qualitative evaluation of HypNO-LWR against classical finite volume baselines and [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: LWR: MAE restricted to the shock band, as a function of the number of discontinuities in the initial [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: LWR: Representative sample for an initial condition partitioned into [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Same LWR representative sample as Fig. 6, zoomed to the shock-band bounding box with non-band [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: ρ(x, t) at t = 0.441 for the widest-band row with an initial condition consisting of N = 2 segments. The shaded region marks the shock band; HypNO-LWR tracks the sharp ground-truth front while the baselines round corners [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: LWR: Representative sample for an initial condition partitioned into [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Same LWR representative sample as Fig. 9, zoomed to the shock-band bounding box with non-band [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: ρ(x, t) at t = 0.504 through the widest band row at N = 5 segments. The shaded region marks the shock band; HypNO-LWR tracks the sharp ground-truth front while the baselines round corners. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: ARZ: ρ and ω - mean absolute error on the ρ and ω fields is plotted agains the initial condition complexity, averaged over the number of samples per bin, for our model, FNO, WENO, HLL and Godunov schemes. Gray area marks segment numbers present in the training set. HypNO-ARZ accuracy stays well below the baseline across the whole complexity range, and it’s error grows more slowly in the out-of-distributio… view at source ↗
Figure 13
Figure 13. Figure 13: In-distribution riemann stratified sample (initial condition partitioned into 2 segments) on the WFT reference: HypNO-ARZ vs. FNO, Godunov and HLL. The learned model keeps the shock and contact sharp, whereas the finite volume baselines diffuse the discontinuities into a multi-cell band. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: In-distribution piecewise constant sample (initial condition partitioned into 5 segments): multiple interacting shocks and contacts. HypNO-ARZ resolves the fronts more sharply than the diffusive Godunov and HLL references, and its error (MAEw = 8.5 × 10−3 ) concentrates on a few thin interfaces rather than spreading across the wave structure. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: ARZ: Out-of-distribution piecewise constant sample (initial condition partitioned into 30 segments, well beyond the training range), evaluated against the wave-front-tracking ground truth. Panels show the density ρ (a) and the Lagrangian marker ω (b), each with predicted space-time fields (top) and mean absolute error to the ground truth (bottom). Despite the interaction density, our model captures the sh… view at source ↗
Figure 16
Figure 16. Figure 16: ARZ: MAE restricted to the shock band, as a function of the number of discontinuities in the initial [PITH_FULL_IMAGE:figures/full_fig_p030_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: ARZ: Representative sample for an initial condition partitioned into [PITH_FULL_IMAGE:figures/full_fig_p031_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Same ARZ representative sample as Fig. 17, zoomed to the combined-band bounding box with non [PITH_FULL_IMAGE:figures/full_fig_p031_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: ρ(x) at t = 0.669 through a band row for ARZ model with num segments=8 initial discontinuity segments. The shaded region marks the combined shock band [PITH_FULL_IMAGE:figures/full_fig_p032_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: ARZ: Representative sample for an initial condition partitioned into [PITH_FULL_IMAGE:figures/full_fig_p032_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Same ARZ representative sample as Fig. 20, zoomed to the combined-band bounding box with non [PITH_FULL_IMAGE:figures/full_fig_p033_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: ρ(x) at t = 0.441 through a band row for ARZ model with num segments=10 initial discontinuity segments. The shaded region marks the combined shock band. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: ARZ: Representative sample for an initial condition partitioned into [PITH_FULL_IMAGE:figures/full_fig_p051_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Same ARZ representative sample as Figure 23, zoomed to the combined-band bounding box with [PITH_FULL_IMAGE:figures/full_fig_p051_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: ρ(x) at t = 0.323 through a band row for ARZ model with 2 initial discontinuity segments. The shaded region marks the combined shock band [PITH_FULL_IMAGE:figures/full_fig_p052_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: ARZ: Representative sample for an initial condition partitioned into [PITH_FULL_IMAGE:figures/full_fig_p052_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Same ARZ representative sample as Fig. 26, zoomed to the combined-band bounding box with non [PITH_FULL_IMAGE:figures/full_fig_p053_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: ρ(x) at t = 0.465 through a band row for ARZ model with 3 initial discontinuity segments. The shaded region marks the combined shock band. 53 [PITH_FULL_IMAGE:figures/full_fig_p053_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: ARZ: Representative sample for an initial condition partitioned into [PITH_FULL_IMAGE:figures/full_fig_p054_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Same ARZ representative sample as Figure 29, zoomed to the combined-band bounding box with [PITH_FULL_IMAGE:figures/full_fig_p054_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: ρ(x) at t = 0.465 through a band row for ARZ model with 5 initial discontinuity segments. The shaded region marks the combined shock band [PITH_FULL_IMAGE:figures/full_fig_p055_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: ARZ: Representative sample for an initial condition partitioned into [PITH_FULL_IMAGE:figures/full_fig_p055_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Same ARZ representative sample as Figure 32, zoomed to the combined-band bounding box with [PITH_FULL_IMAGE:figures/full_fig_p056_33.png] view at source ↗
Figure 34
Figure 34. Figure 34: ρ(x) at t = 0.496 through a band row for ARZ model with 7 initial discontinuity segments. The shaded region marks the combined shock band. 56 [PITH_FULL_IMAGE:figures/full_fig_p056_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: ARZ: Representative sample for an initial condition partitioned into [PITH_FULL_IMAGE:figures/full_fig_p057_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: Same ARZ representative sample as Figure 35, zoomed to the combined-band bounding box with [PITH_FULL_IMAGE:figures/full_fig_p057_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: ρ(x) at t = 0.283 through a band row for ARZ model with 20 initial discontinuity segments. The shaded region marks the combined shock band [PITH_FULL_IMAGE:figures/full_fig_p058_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: ARZ: Representative sample for an initial condition partitioned into [PITH_FULL_IMAGE:figures/full_fig_p058_38.png] view at source ↗
Figure 39
Figure 39. Figure 39: Same ARZ representative representative sample as Figure 38, zoomed to the combined-band bounding [PITH_FULL_IMAGE:figures/full_fig_p059_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: ρ(x) at t = 0.173 through a band row for ARZ model with 30 initial discontinuity segments. The shaded region marks the combined shock band. 59 [PITH_FULL_IMAGE:figures/full_fig_p059_40.png] view at source ↗

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