{"id":"792fba34-c5a4-43c6-b679-b5476e34cfc4","arxiv_id":"2506.22413","paper_version":3,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":3.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"The work investigates the sensitivity of PINNs to conservative versus non-conservative PDE formulations when solving benchmark problems that contain shocks and discontinuities.","lead":"This paper tests Physics-Informed Neural Networks on conservative versus non-conservative forms of PDEs with shocks, using Burgers and Euler equations as benchmarks. A smart generalist might read it to learn whether neural solvers can avoid the need to rewrite many physical equations into conservative form for accurate discontinuity capture.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"Benchmarks limited to Burgers/Euler may not represent challenges of inherently non-conservative PDEs (e.g. multi-phase flows)","rationale":"The reader's weakest_assumption directly identifies the same load-bearing gap between the stated motivation and the chosen test suite. Because the abstract explicitly lists only Burgers and Euler, and no independent verification (e.g., machine-checked proofs or additional multi-phase results) is mentioned, the concern stands without modification to the UNVERDICTED verdict.","tokens_in":1674,"tokens_out":335,"duration_ms":33719,"concrete_test":"Apply the same PINN architecture and loss weighting to a non-conservative two-fluid model (e.g., Baer-Nunziato or a simplified two-phase Euler with non-conservative interfacial terms) containing a shock; compare L2 error and shock smearing against the conservative-form counterpart. If the relative error gap between formulations exceeds that observed for Euler, the benchmark limitation is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the chosen benchmarks (Burgers equation, steady/unsteady Euler) yield a comprehensive understanding of PINN sensitivity to conservative vs. non-conservative formulations near shocks. For this to support the motivating application to inherently non-conservative systems (multi-phase flows with non-conservative products that cannot be rewritten in divergence form), the test problems must capture the same structural difficulties. Burgers and Euler admit both forms and remain hyperbolic conservation laws; they do not test cases where non-conservativity is intrinsic and the conservative form is unavailable. Without such cases the sensitivity results remain conditional on the benchmark class.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1782,"tokens_out":333,"duration_ms":11081,"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":[{"comment":"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).","section":"Abstract / Benchmark selection"}],"minor_comments":[{"comment":"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.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their thoughtful comments. We address the major comment below and indicate where revisions will be made.","responses":[{"response":"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_made":"yes","referee_comment":"[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)."}],"tokens_in":1236,"tokens_out":276,"duration_ms":23251,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper runs PINNs on Burgers and steady/unsteady Euler to see if the network output changes when you switch between the two PDE forms near discontinuities. That is the core activity: no new architecture or loss term, just a side-by-side test on problems that already have both formulations available in the literature.","headline":"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.","tokens_in":2271,"tokens_out":140,"would_cite":false,"duration_ms":19813,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":{"model":"grok-4.3","evidence":[],"headline":"PINN sensitivity study for conservative/non-conservative PDEs shows no RS-shaped machinery","alignment":"orthogonal","rationale":"Paper centers on empirical behavior of adaptive-weight/viscosity PINNs on Burgers and Euler equations (conservative vs. non-conservative forms, shock speed, artificial viscosity). No J-cost, ratio symmetry, φ-ladder, 8-tick periodicity, or parameter-free constant derivations appear. Benchmarks and loss formulations are standard CFD/ML and do not parallel any RS theorem (e.g., reality_from_one_distinction, J_uniquely_calibrated_via_higher_derivative, alexander_duality_circle_linking).","tokens_in":53455,"confidence":"high","tokens_out":153,"duration_ms":5461,"cache_read_input_tokens":38528,"cache_creation_input_tokens":0},"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Physics-informed neural networks perform similarly on shocks whether equations are written in conservative or non-conservative form.","keywords":["physics-informed neural networks","conservative form","non-conservative form","shocks","discontinuities","Burgers equation","Euler equations","computational fluid dynamics"],"falsifier":"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.","tokens_in":2591,"feed_emoji":"","tokens_out":594,"duration_ms":23762,"temperature":0.7,"pith_summary":"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.","feed_headline":"PINNs solve shocks in either equation form","feed_subtitle":"Tests on Burgers and Euler equations show limited sensitivity to conservative versus non-conservative formulations for capturing flow jumps.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["PINNs show limited sensitivity to PDE form in shock capturing","Burgers Euler tests confirm PINNs work with non-conservative equations","No major difference for PINNs using conservative or non-conservative formulations","PINNs capture discontinuities irrespective of conservation form","PINNs insensitive to conservation form in solving flows with shocks"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The chosen benchmark problems sufficiently represent the numerical challenges posed by inherently non-conservative PDEs in complex physical phenomena such as multi-phase flows.","fun_headline_variants_meta":{"raw":{"variants":["PINNs show limited sensitivity to PDE form in shock capturing","Burgers Euler tests confirm PINNs work with non-conservative equations","No major difference for PINNs using conservative or non-conservative formulations","PINNs capture discontinuities irrespective of conservation form","PINNs insensitive to conservation form in solving flows with shocks"]},"model":"grok-4.3","cost_usd":0.004446,"raw_usage":{"total_tokens":2180,"prompt_tokens":588,"num_sources_used":0,"completion_tokens":80,"cost_in_usd_ticks":44462000,"prompt_tokens_details":{"text_tokens":588,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1512,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":588,"tokens_out":80,"duration_ms":15658,"temperature":1.0,"reasoning_tokens":1512,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-05-25T08:11:19.386811+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}