{"id":"d17e2335-9310-47a3-bf26-3ee0764489be","arxiv_id":"2506.11959","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"A PINN variant that applies a Roe-type approximate Riemann solver only at detected shock locations produces sharper, more conservative shock solutions in hydrodynamics and relativistic hydrodynamics without needing shock speeds in advance.","lead":"The authors modify how physics-informed neural networks handle shock waves by borrowing a trick from classical shock-capturing schemes: locally replacing the equations' Jacobian with a Roe-averaged version only where the flow compresses. The result is sharper shocks and better conservation in 1D and 2D test problems, but small vortex structures are still smeared.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Roe property (50) is pointwise, not weak-form; LRPINN loss does not transfer it to RH conditions, and Fig. 7 shows only ~1e-3 conservation.","rationale":"The reader's weakest assumption is precisely the load-bearing gap: the paper proves the algebraic Roe property but not that pointwise residual minimization with a modified Jacobian yields a weak solution. My stress-test confirms this is not merely a missing formality. The residual in Algorithm 2 is evaluated at single collocation points, whereas the Rankine-Hugoniot condition is a relation between separated left and right states across a discontinuity. The network is smooth, so it cannot represent the jump in the same way a finite-volume interface does; the Roe-averaged matrix built from two nearby points on a smooth profile is close to the local Jacobian, meaning the modification is weak until a near-discontinuity forms, and there is no mechanism that converts the pointwise residual into an integral conservation statement. The paper's own Figure 7 shows conservation error of order 10^-3 for LRPINN, which is good engineering performance but not 'ensuring' jump conditions. This does not undermine the empirical usefulness of the method; the error tables against WENO are credible evidence of practical accuracy. However, the conclusion overstates the theoretical guarantee. The appropriate verdict remains CONDITIONAL: the method is promising and well tested on benchmarks, but the central claim should be softened to 'approximately enforces' or supported by a weak-form residual analysis. My concern does not change the reader's verdict, so verdict_should_be is UNCHANGED.","tokens_in":24857,"tokens_out":3662,"duration_ms":41029,"concrete_test":"Take the trained LRPINN for Problem 3 (Lax tube) and compute the weak-form residual: for a set of smooth compactly supported test functions φ on the space-time domain, evaluate W(φ)=∫[U ∂_t φ + F(U) ∂_x φ] dxdt + ∫ U(x,0)φ(x,0)dx. If max_φ |W(φ)| is not orders of magnitude below the pointwise modified-residual mean, then the Roe property does not transfer to the weak solution, and the 'ensures jump conditions' claim should be downgraded. Also report the L1 deviation of the shock jump from Eq. (22) using states at x_s±Δx.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that LRPINNs 'robustly ensure' jump conditions is not supported by the argument in Sec. III.C. The Roe property (50) is an algebraic identity for constant left/right states across a discontinuity: F(U_R)-F(U_L)=A_roe(U_R-U_L). But Algorithm 2 uses A_roe only to evaluate the pointwise residual ∂U/∂t + A_roe(U_L,U_R) ∂U/∂x at each detected compression point. Minimizing the sum of squares of these pointwise residuals does not minimize any weak-form residual; a smooth network can satisfy the modified quasi-linear equation without satisfying the conservation law's integral form. Moreover, U_L and U_R are evaluated at x_s±Δx on the network's smooth profile, not at a true discontinuity; the Roe average of two nearby smooth states is essentially the local Jacobian, so the mechanism does not act as a Riemann solver at a cell interface. The paper's own global conservation check (Fig. 7) shows LRPINN error ~10^-3, not machine precision, contradicting the 'ensures' language. No proof connects residual minimization to RH satisfaction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes two modifications of locally linearized PINNs (LLPINNs) for hyperbolic conservation laws with shocks. First, a generalized LLPINN computes shock speeds from neighboring network states via Rankine-Hugoniot formulas and uses Lax entropy conditions to select which eigenvalue to replace, removing the need to prescribe shock velocities in advance. Second, LRPINNs replace the Jacobian in detected compression regions by a Roe-averaged Jacobian, aiming to enforce jump conditions and improve conservation. Both methods are extended to the 2D Euler equations using a divergence-based shock detector and direction-wise Jacobian modification. The authors test the methods on 1D Euler shock-tube problems, 1D relativistic hydrodynamics, and 2D Riemann problems, comparing against exact solutions, WENO-Z, and HLLC solvers. They report sharper shock resolution and better conservation than classical PINNs and LLPINNs, with LRPINN accuracy comparable to or better than WENO at moderate resolution.","tokens_in":25183,"tokens_out":5138,"duration_ms":68309,"significance":"If the central claims were fully established, the paper would make a useful contribution: it would provide a practical way to embed approximate Riemann solver information into PINNs without a priori shock speeds, and the numerical comparisons show consistent gains over standard PINNs and the original LLPINN. The detailed hyperparameter tables and algorithm listings are a strength, and the inclusion of relativistic and 2D problems broadens applicability. However, the central theoretical claim that jump conditions are 'ensured' is not proven; the manuscript's own conservation diagnostic shows only approximate conservation. The contribution is therefore better characterized as an empirical method with heuristic motivation, whose claims need to be scaled back or supplemented with a weak-form consistency argument.","major_comments":[{"comment":"The argument that LRPINNs 'robustly ensure' jump conditions is not established. Eq. (50) is an algebraic identity for constant states U_L and U_R, but Algorithm 2 only uses the Roe-averaged Jacobian inside the pointwise residual D[U] = ∂U/∂t + A_roe ∂U/∂x evaluated at collocation points. Minimizing the sum of squared pointwise residuals does not minimize any weak-form residual, and a smooth network can satisfy the modified quasi-linear system without satisfying the integral conservation law. The paper's own conservation check in Fig. 7 shows a residual of order 1e-3, not machine precision, so the 'ensures' language is not supported. Please either add a proof or argument connecting residual minimization to Rankine-Hugoniot satisfaction, augment the loss with an explicit weak-form or integral conservation term, or soften the claims to 'encourages' or 'approximately enforces'.","section":"Section III.C, Eq. (50), and Algorithm 2"},{"comment":"The left and right states are obtained by evaluating the network at x_s ± Δx, where Δx is a hyperparameter. Since the network is smooth and Δx is small (Table I, values around 0.001 to 0.02), U_L and U_R are not the two branches of a discontinuous Riemann state; in the limit Δx→0 their Roe average tends to A(U(x_s)), so the modification degenerates to the local Jacobian. The paper should justify that these neighboring values are meaningful Riemann states for the shock and report sensitivity to Δx; otherwise the mechanism is not demonstrably acting as a Riemann solver at an interface.","section":"Algorithm 2, steps 7-8"},{"comment":"The statement that the generalized LLPINN 'ensures that the Rankine–Hugoniot relations (22) and entropy conditions (23) are satisfied by construction' is an overstatement. Algorithm 1 replaces an eigenvalue by s_i in the quasi-linear residual, which encodes the shock speed in the differential operator but does not by itself enforce the jump condition in the network output. The conclusion should be rephrased to describe the intended design and the observed empirical behavior rather than a guaranteed property.","section":"Section V, Conclusions"}],"minor_comments":[{"comment":"The problem numbering is inconsistent: 'Problem 4' appears twice (Sod shock tube in a moving reference frame, and relativistic Sod shock tube), and 'Problem 5' appears twice (relativistic double shock wave and, implicitly, the next problem). Please renumber the 1D and 2D problems sequentially.","section":"Section IV.A, Problem numbering"},{"comment":"The shock-speed formulas appear to be missing a square-root symbol and closing parentheses: the expressions should read s_- = u_L - a_L * sqrt( (1/(2γ)) * ((γ+1) p_R/p_L + γ-1) ) and similarly for s_+. Please check the typesetting.","section":"Eqs. (31)-(32)"},{"comment":"There are small typographical errors in the residual definitions: Algorithm 3 line 12 and Algorithm 4 line 11 contain a double plus sign (∂U/∂t + Ã ∂U/∂x + + B̃ ∂U/∂y), and Algorithm 3 step 5 writes '∂_y v' without the ∂ symbol. These should be corrected.","section":"Algorithm 3 and Algorithm 4"},{"comment":"The abstract says the method 'applies jump conditions through entropy constraints,' but the algorithms use Lax entropy conditions only to select which eigenvalue to modify; the jump conditions enter through the Rankine-Hugoniot shock-speed formulas. Please clarify this distinction.","section":"Abstract and Section V"},{"comment":"The caption says 'conserved quantities' but does not specify which component or combination is plotted, nor whether the difference is normalized. Please state whether the plotted quantity is mass, momentum, energy, or a composite, and clarify the normalization.","section":"Figure 7"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the skeptical concern raised in the reader's report is valid and lands on the central claim. The paper's empirical material is useful and the method may well be a practical improvement, but the headline claim that LRPINNs 'ensure' jump conditions is not supported by the presented analysis. I would not reject, because the issue can be addressed by either adding a weak-form or integral conservation loss, providing a consistency argument, or softening the wording. The duplicated problem numbers and several typos should also be fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid incremental advance. The genuinely new pieces are three: a generalized LLPINN that estimates shock speeds from neighboring states instead of requiring them as input, LRPINNs that replace the Jacobian with the Roe-averaged Jacobian inside detected compression regions, and a divergence-based dimensional-splitting extension to 2D. The Roe-averaged Jacobian is new to PINN training, and the 2D detection is a sensible extension. The 1D results are convincing: LRPINNs resolve shocks and contacts far better than vanilla PINNs, and the quantitative comparison against WENO-Z in Table II shows LRPINN accuracy comparable to a fine-grid WENO solution. That is an honest and useful benchmark.\n\nThe main soft spot is the paper's load-bearing claim that LRPINNs 'robustly ensure that both shock velocities and jump conditions are met.' That is not proven. Equation (50) is a pointwise algebraic identity for the Roe matrix; using that matrix in the quasi-linear residual does not by itself transfer the Roe property to the weak form of the conservation law. Minimizing this residual is not minimizing a weak-form residual. The paper's own conservation check (Fig. 7) shows error around 1e-3, not machine precision, which contradicts the 'ensures' language. This is an overstatement, not a fatal flaw—the method can still be a useful heuristic that improves conservation—but the claims should be pulled back to something like 'promotes' or 'encourages.'\n\nSecond soft spot: the 2D validation is almost entirely visual. No error norms are reported for the 2D Riemann problems, and the acknowledged smoothing of small-scale vortex structures is not quantified. The method also relies on several per-problem hyperparameters (M, Mx, My, Delta x, viscosity, etc.) with no sensitivity analysis and no released code, which limits reproducibility.\n\nWho is this for? Researchers working on PINNs for hyperbolic conservation laws, especially those who want shock-capturing behavior without giving up the mesh-free PINN framework. The paper deserves a serious referee. With the overclaim corrected and some 2D error metrics added, it would be a good contribution. I would send it to review.","headline":"A genuinely useful PINN tweak for shocked flows, with an overclaimed guarantee that should be softened before publication.","tokens_in":25650,"tokens_out":1780,"would_cite":true,"duration_ms":116610,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35L65","65M06","68T07"],"pacs":["47.40.-i"],"model":"deepseek-v4-flash","headline":"Roe-averaged Jacobians put shock jump conditions into PINN training","keywords":["physics-informed neural networks","Riemann solvers","hyperbolic conservation laws","Rankine-Hugoniot conditions","Roe averaging","shock capturing","relativistic hydrodynamics","Euler equations"],"falsifier":"On the Sod or Lax shock tube, run the LRPINN with the settings of Algorithm 2 and track $\\int (U_{\\mathrm{pred}}(x,t)-U_{\\mathrm{pred}}(x,0))\\,dx$ over time; if this conservation residual grows like the LLPINN curve in Figure 7 instead of staying near $10^{-3}$, the claim that the Roe replacement enforces jump conditions fails. Alternatively, evaluate the trained network on a fine uniform grid across the shock and check whether $F(U_R)-F(U_L) \\approx s(U_R-U_L)$ with the predicted $s$; a clear violation indicates the property did not survive training.","tokens_in":24620,"feed_emoji":"💥","tokens_out":4311,"duration_ms":48155,"temperature":0.7,"pith_summary":"The paper tries to establish that a small change inside a PINN's residual—swapping the Jacobian matrix for the Roe-averaged Jacobian only in detected compression regions—makes the network honor shock jump conditions without knowing shock speeds in advance. The authors generalize the earlier locally linearized PINNs by computing shock speeds from neighboring states with entropy conditions, and introduce LRPINNs built on Roe's approximate Riemann solver. They report that LRPINNs keep global conservation residuals nearly flat over time, reproduce sharp shocks in one-dimensional Euler and relativistic hydrodynamics, and extend to two-dimensional Riemann problems via divergence-based shock detection and dimensional splitting. A sympathetic reader cares because the method offers a path to using PINNs on genuinely discontinuous flows with conservation guarantees, matching or beating a high-order WENO solver on the benchmarked problems.","feed_headline":"Roe-averaged Jacobians put shock jump conditions into PINN training","feed_subtitle":"A locally Roe-averaged PINN meets Rankine-Hugoniot conditions without knowing shock speeds in advance.","key_machinery":"The Roe-averaged Jacobian matrix $\\tilde A$, defined by density-weighted averages $\\tilde u$, $\\tilde v$, $\\tilde H$ (and the Eulderink–Mellema averaged state for relativistic hydrodynamics), satisfies the jump identity $F(U_R)-F(U_L)=\\tilde A(U_R-U_L)$. The network's PDE residual $\\partial_t U + \\tilde A\\,\\partial_x U=0$ is computed with this modified matrix only inside the compression set $\\Omega_s$ defined by a negative velocity gradient or negative divergence, so the shock-detection criterion decides where the modification acts. This machinery transfers the approximate Riemann solver's conservation property into the loss function.","core_discovery":"The central discovery is that enforcing the Roe property $F(U_R)-F(U_L)=\\tilde A(U_R-U_L)$ locally, through the Roe-averaged Jacobian, is enough to turn the quasi-linear residual into a device that carries the jump conditions into the network training. Standard PINNs, even with viscous pretraining, mislocate shocks and lose contact waves; LLPINNs correctly predict shock speeds but violate conservation because their eigenvalue-substituted Jacobian does not satisfy the Roe property. LRPINNs replace the Jacobian only in regions where $\\partial u/\\partial x \\le -M$ (or, in two dimensions, where $\\nabla\\cdot u \\le -M$), with the Roe matrix built from neighbor states, and leave the true Jacobian elsewhere. The authors show that this yields accurate shock locations, sharp transitions, and conserved mass, momentum, and energy over time, and extends to two dimensions by applying the modification direction-by-direction with dimensional splitting.","pith_inferences":["The same selectively Roe-averaged residual principle could be transferred to other PINN formulations for conservation laws with available approximate Riemann solvers, such as shallow-water or magnetohydrodynamics, where Roe-type matrices exist; the authors gesture at this but do not test it.","Because the shock speed emerges from the Roe property rather than being an input, LRPINNs may be useful for inverse problems where shock trajectory or wave speed is the quantity to recover; that use is not explored here.","The sharper-shock-but-smoother-vortex trade-off observed in two dimensions suggests that a hybrid loss, mixing the Roe-modified residual with a spectral or multiscale term, could recover small-scale features; this is a testable design not addressed by the paper.","A direct comparison of conservation residual against a fully conservative PINN formulation, such as integral-form residuals, would clarify whether the local Roe replacement or the quasi-linear structure is the active ingredient."],"forward_implications":["LRPINNs solve one-dimensional Riemann problems for Euler and relativistic Euler equations without any prior knowledge of shock speeds, matching or exceeding WENO-Z accuracy up to about 1600 grid points in the tested double-shock case.","Global conservation of mass, momentum, and energy stays approximately constant over time, unlike LLPINNs where conservation residuals grow with time (Figure 7).","The method correctly handles transonic rarefactions in a moving-frame Sod tube because the Roe replacement is limited to compression regions, avoiding the entropy-violating behavior of pure Roe solvers.","In two-dimensional Riemann problems, divergence-based detection plus dimensional splitting yields sharp shock and contact transitions while maintaining cylindrical symmetry (Problem 6).","The approach is not tied to Roe specifically: the authors state that other approximate solvers or the exact Riemann solver could be substituted by replacing eigenvalues and reconstructing the modified Jacobian."],"supporting_citations":[{"why":"Supplies the original LLPINN method with eigenvalue substitution that this paper generalizes and whose conservation failure it diagnoses.","marker":"[36]"},{"why":"Supplies the Roe-averaged Jacobian and the jump property (50) that define the LRPINN modification.","marker":"[37]"},{"why":"Supplies the gradient-annihilation factor and the relativistic benchmark problems used as test cases.","marker":"[35]"},{"why":"Supplies the relativistic Roe-averaged state used in the special-relativistic LRPINN implementation.","marker":"[78]"},{"why":"Supplies the Rankine-Hugoniot conditions, Lax entropy conditions, and dimensional splitting framework underpinning the algorithms.","marker":"[9]"},{"why":"Supplies the high-order WENO-Z solver used as the reference for one-dimensional error comparisons.","marker":"[81]"},{"why":"Supplies the two-dimensional Riemann problem configurations used as benchmarks for the multidimensional extension.","marker":"[89]"}],"fun_headline_variants":["Roe-averaged Jacobians let PINNs capture shocks without knowing speeds beforehand","Locally Roe-averaged PINNs fix shock jump conditions and conserve mass, momentum","LRPINNs: Roe solver inside neural net yields sharp shocks and conserved flows","PINN with Roe-averaged Jacobian: shock speeds emerge, not prescribed"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that training on the quasi-linear residual with the Roe Jacobian in compression regions makes the network output satisfy the weak conservation law at the shock; the paper proves the Roe matrix has the jump property, but does not prove that residual minimization transfers this property to the network output.","fun_headline_variants_meta":{"raw":{"variants":["Roe-averaged Jacobians let PINNs capture shocks without knowing speeds beforehand","Locally Roe-averaged PINNs fix shock jump conditions and conserve mass, momentum","LRPINNs: Roe solver inside neural net yields sharp shocks and conserved flows","PINN with Roe-averaged Jacobian: shock speeds emerge, not prescribed"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000323,"raw_usage":{"total_tokens":1828,"prompt_tokens":972,"completion_tokens":856,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":769}},"tokens_in":588,"tokens_out":856,"duration_ms":9700,"temperature":1.0,"reasoning_tokens":769,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:59:50.052342+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the Sod or Lax shock tube, run the LRPINN with the settings of Algorithm 2 and track $\\int (U_{\\mathrm{pred}}(x,t)-U_{\\mathrm{pred}}(x,0))\\,dx$ over time; if this conservation residual grows like the LLPINN curve in Figure 7 instead of staying near $10^{-3}$, the claim that the Roe replacement enforces jump conditions fails. Alternatively, evaluate the trained network on a fine uniform grid across the shock and check whether $F(U_R)-F(U_L) \\approx s(U_R-U_L)$ with the predicted $s$; a clear violation indicates the property did not survive training.","supporting_citations":[{"cited_title":"Michoski , author M","cited_arxiv_id":null,"evidence_quote":"Supplies the original LLPINN method with eigenvalue substitution that this paper generalizes and whose conservation failure it diagnoses."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the Roe-averaged Jacobian and the jump property (50) that define the LRPINN modification."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the relativistic Roe-averaged state used in the special-relativistic LRPINN implementation."},{"cited_title":"General Relativistic Hydrodynamics with a Roe solver","cited_arxiv_id":"astro-ph/9411056","evidence_quote":"Supplies the high-order WENO-Z solver used as the reference for one-dimensional error comparisons."},{"cited_title":"Wang , author H","cited_arxiv_id":null,"evidence_quote":"Supplies the two-dimensional Riemann problem configurations used as benchmarks for the multidimensional extension."}],"review_version":1}