{"id":"d639657a-052c-4015-aff1-ac5e18fdb376","arxiv_id":"2507.09757","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"An energy-dissipation-rate-guided sampling strategy (EDRAS) improves PINN accuracy on thermodynamically consistent Allen-Cahn models, giving up to a sixfold relative MSE reduction over residual-based adaptive refinement in a 1D test.","lead":"EDRAS is a new adaptive sampling strategy for physics-informed neural networks that chooses collocation points where the physical energy is being dissipated fastest, instead of where the PDE residual is largest. It is tested on Allen-Cahn phase-field equations in 1D and 2D irregular domains, where it reports up to a sixfold error reduction over residual-based RAR sampling in the 1D benchmark.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Algorithm 1 defines the EDRAS sampling metric as signed phi_t/M_b, contradicting the nonnegative dissipation density phi_t^2/M_b required by Eqs. (3) and (11); the method as written is not reproducible.","rationale":"I read the paper in good faith. The central claim is that sampling guided by energy dissipation rate density improves PINN accuracy over residual-based RAR for thermodynamically consistent Allen-Cahn systems. The 1D numerical evidence is clear and the physical motivation is plausible, but the strongest claim rests on a metric that the paper defines inconsistently. The reader's weakest-assumption analysis focused on whether dissipation density correlates with approximation error; that is a legitimate concern. My stress-test identifies a more concrete, load-bearing problem: the actual formula in Algorithm 1 is not the dissipation rate density from Eq. (3) or Eq. (11). Because all of the reported comparisons, including the probability plots in Figure 7 and the headline sixfold MSE improvement, are produced by this sampling procedure, the results cannot be fully verified or reproduced from the manuscript as written. The issue is fixable by correcting the formula and rerunning the experiments, so I do not recommend rejection. I retain the reader's conditional verdict, but the condition should explicitly require clarification and numerical confirmation of the exact EDRAS metric. The reader did not flag this internal formula contradiction, so my agreement is partial.","tokens_in":17558,"tokens_out":5092,"duration_ms":59282,"concrete_test":"Rerun the 1D Allen-Cahn experiment in Eq. (10) with exactly the same initialization, network architecture, and optimizer settings three times: once with eedrdf = phi_t/M_b as written in Algorithm 1, once with eedrdf = |phi_t|/M_b, and once with eedrdf = phi_t^2/M_b. Compare the resampled point distributions in [0.8,1] x [-1,1] and the Table 1 error metrics. If the relative MSE or the selected point sets differ materially (say, by more than 20% in relative MSE, or visibly different spatial clustering), then the reported EDRAS advantage must be re-reported under a single corrected definition, and the probability analysis in Figure 7 must be regenerated accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central mechanism is the use of local energy dissipation rate density to guide resampling, but Algorithm 1's definition of that density is internally inconsistent with the model's thermodynamics. Eq. (3) gives the bulk dissipation rate integrand as -phi_t^2/M_b, and the 1D expression in Eq. (11) is -phi_t^2/5; the text says eedrdf is the absolute value of this density, which should be phi_t^2/M_b (or |phi_t|/M_b, if only the magnitude of the flux is meant). Algorithm 1, step 3, however, states eedrdf = phi_t/M_b and removes points with eedrdf < eedrdf0, where eedrdf0 = (1/(3|J_f|)) sum eedrdf. Since phi_t changes sign during Allen-Cahn evolution, this signed quantity is not a dissipation rate density: points with negative phi_t are discarded regardless of how large |phi_t| is, while regions with small positive phi_t can be retained. The threshold-based removal in step 3 and the top-m addition in step 8 are both sensitive to the sign or square, so the resampled point sets shown in Figures 5 and 7, and hence the reported sixfold MSE improvement in Table 1, depend on which of phi_t/M_b, |phi_t|/M_b, or phi_t^2/M_b was actually implemented. The manuscript never resolves this, and no code is provided. This is not a mere notation quibble: the method's name and all of its claimed advantages rest on sampling regions of high dissipation, yet the algorithm as printed can select entirely different regions.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces EDRAS, an adaptive collocation-point sampling strategy for PINNs that uses the local energy dissipation rate density of a thermodynamically consistent model as the guide for adding and removing training points, instead of the PDE residual. The method is applied to the Allen-Cahn equation in 1D with periodic boundary conditions and in 2D disk/ellipse domains with Neumann or dynamic boundary conditions. The authors report up to a sixfold reduction in relative MSE compared with residual-based adaptive refinement (RAR) in the 1D benchmark, provide a probability-based argument for why RAR can miss low-residual/high-error regions, and draw qualitative conclusions about how surface mobility affects bulk dynamics. The 1D and 2D Neumann cases are validated against finite-difference reference solutions; the dynamic-boundary cases are not.","tokens_in":17897,"tokens_out":10660,"duration_ms":119774,"significance":"If the central claim is established, EDRAS would be a useful and computationally cheap alternative to residual-based adaptive sampling for dissipative PDEs, because it replaces higher-order residual derivatives with a first-order dissipation-rate density and requires no global density estimation. The 1D benchmark is clean and uses an external FDM reference, which avoids circularity, and the probability-based comparison in Figures 6-7 is a helpful conceptual explanation of a known limitation of RAR. However, the evidence for superiority over RAR is currently limited to one 1D problem, the printed algorithm is internally inconsistent in its definition of the sampling metric, and the physical insights from the dynamic-boundary cases are not independently validated. The method also has several under-specified hyperparameters. These issues are fixable, and the underlying idea is promising.","major_comments":[{"comment":"The definition of the EDRAS metric in Algorithm 1 is inconsistent with the model's thermodynamics. Eq. (3) gives the bulk dissipation-rate integrand as -phi_t^2/M_b (and Eq. (11) as -phi_t^2/5), and Section 2.2 states that eedrdf is the absolute value of the energy dissipation-rate density, so eedrdf should be phi_t^2/M_b or at least |phi_t|/M_b. Algorithm 1 instead sets eedrdf = phi_t/M_b and uses this signed quantity in the thresholding (step 3) and in the top-m selection (step 8). Since phi_t changes sign during Allen-Cahn evolution, this is not a dissipation-rate density: regions with large negative phi_t are removed regardless of magnitude, while regions with small positive phi_t can be retained. The threshold eedrdf0 can itself become negative under this definition. The resampled point sets that produce the reported results in Table 1 and Figures 5 and 7 therefore depend on which of phi_t/M_b, |phi_t|/M_b, or phi_t^2/M_b was actually implemented. Please correct the algorithm to match the thermodynamics and state which quantity was used in each experiment; without this, the central numerical claim is not reproducible from the printed method.","section":"Algorithm 1, steps 2-3 and 8; Eq. (3); Eq. (11); Section 2.2"},{"comment":"The paper's headline claim is that EDRAS outperforms RAR, but the only head-to-head quantitative comparison is the 1D periodic problem in Table 1. In the 2D Neumann case, Section 3.2 says adaptive refinement was performed with EDRAS and RAR methods independently or in combination, yet Table 2 and Figure 8 report only EDRAS errors against the FDM reference; no RAR-only or combined 2D error metrics are shown. The 2D dynamic-boundary studies in Section 3.3 contain no reference solution and no RAR comparison. Consequently, the abstract's implication that EDRAS improves PINNs in arbitrary domains relative to RAR is supported only by a single 1D benchmark. Please add at least one 2D comparison, such as the disk with Neumann boundary conditions, under identical training settings, with error metrics for RAR, EDRAS, and EDRAS+RAR.","section":"Section 3.2, Table 2; Section 3.3; Abstract"},{"comment":"The physical conclusions about dynamic boundary conditions, e.g., that higher surface mobility accelerates boundary coarsening and slightly accelerates bulk energy decay, are drawn exclusively from PINN solutions with dynamic boundary conditions. No independent reference solution or error estimate is provided for these cases; the only external 2D validation is for the Neumann case (Table 2). Because EDRAS is a heuristic sampling strategy, the observed differences between Ms=10 and Ms=2 or 5 could lie within the approximation error of the PINN. Please validate at least one dynamic-boundary configuration against an FDM or other grid-converged reference, or explicitly rephrase these conclusions as qualitative observations that are not quantitatively established.","section":"Section 3.3, Figures 10-13; Section 3.2"},{"comment":"The algorithm depends on the number of subdomains N, the density thresholds d_f^(0) and d_b^(0), and the 1/3 factor in the dissipation threshold. The manuscript never states the value of N used in the experiments, and the description of d_f^(0) as being determined from the total number of points in a uniform grid is not a precise recipe in a mesh-free PINN setting. The 1/3 factor is introduced without justification or sensitivity analysis; Table 3 varies d_f^(0) but not the 1/3 factor or N. These choices directly affect which points are added and removed, so the method as printed is not fully reproducible. Please provide the exact parameter values used in each numerical experiment and include a brief sensitivity discussion for the 1/3 factor.","section":"Algorithm 1; Section 2.2; Section 3.2, Table 3"}],"minor_comments":[{"comment":"The caption states that 'EDRAS focuses exclusively on the error', but EDRAS selects points using the energy dissipation-rate density, which the paper argues is a proxy for error; the wording should be corrected to avoid implying that the error itself is used in the sampling criterion.","section":"Figure 7 caption, Section 3.1"},{"comment":"The statement pRAR(B\\cup D)=1 is a conditional conclusion that holds only when m is small enough that every selected top-m residual exceeds the threshold R0; the manuscript should state this condition explicitly rather than presenting it as unconditional.","section":"Section 3.1, probability discussion"},{"comment":"The dissipation-rate density is written with a leading minus inside the integrand; writing E'(t) = -\\int_\\Omega phi_t^2/M_b dx - \\int_{\\partial\\Omega} phi_t^2/M_s dS would avoid any ambiguity about the sign of phi_t^2.","section":"Eq. (3) and Eq. (18)"},{"comment":"The phrase 'density of bulk energy dissipation rate density' is redundant; consider using 'bulk energy dissipation-rate density' throughout the algorithm.","section":"Algorithm 1, step 3"},{"comment":"The statement that no data were used is surprising given the extensive numerical experiments; if the simulations are reproducible from the described code, the authors should state that simulation data are available on request or deposit the code.","section":"Data Availability"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of math.NA and the EDRAS concept is attractive. The main barrier is not the heuristic nature of the method but the inconsistency in Algorithm 1 and the lack of a 2D RAR comparison; both are fixable. The dynamic-boundary physical claims are also asserted without independent validation, which is a concern given that one of the paper's advertised contributions is the physical insight. If the authors resolve these points, the paper would be a reasonable candidate for publication; otherwise the central performance claim rests on a single 1D example and an ambiguous algorithm."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the energy-dissipation-rate-guided sampling idea is new, physically motivated, and the 1D comparison against RAR is the strongest part of the paper. But Algorithm 1 defines the sampling metric as signed phi_t/M_b, which is not the dissipation rate density from Eq. (3), and the text calls for the absolute value or the square. That is a load-bearing inconsistency: the set of resampled points, and hence the sixfold MSE numbers, depends on which definition was actually used. No code is provided, so the method as written is not reproducible.\n\nWhat the paper does well: the motivation is clear and honest. The analogy to condition-number issues in GMRES, and the four-group probability analysis of RAR vs RAD, explains a real failure mode of residual-based sampling in Allen-Cahn. The dissipation-rate density from the FDM reference aligns visually with solution error in Figure 2, which supports the proxy. The energy identities (3) and (11) are correct. The idea of using lower-order derivatives (flux phi_t instead of residual with phi_xx) for computational savings is reasonable and worth testing.\n\nSoft spots, in proportion: the Algorithm 1 inconsistency is the main one. The text says eedrdf is the absolute value of the dissipation density, and Eq. (3) gives phi_t^2/M_b, but Algorithm 1 step 3 uses phi_t/M_b without absolute value or square. Since phi_t changes sign, this can prune the exact regions the method is supposed to keep. This has to be resolved before the numerical claims can be fully trusted. The 1/3 threshold in eedrdf0 is arbitrary, and only d_f^0 gets a sensitivity study; the appendix/table does not vary the 1/3 fraction or the number of subdomains N. In 2D, the Neumann case is validated against FDM, but the dynamic-boundary cases are not validated against any independent reference, so the physical insights about surface mobility effects on bulk dynamics are suggestive, not established. Also, the paper says EDRAS 'focuses exclusively on the error' in Figure 7, but dissipation density is a proxy for error, not error itself; the empirical correlation is one example, not a guarantee.\n\nWho it is for: people working on PINN adaptive sampling, and anyone solving thermodynamically consistent PDEs with PINNs. The core idea is worth a serious referee. Send it out, with a request to fix the algorithm definition, add a reproducibility statement or code, and add at least one 2D RAR comparison with dynamic BCs. I would not cite it in its current form, but I would read the revision.","headline":"A genuinely new PINN sampling idea with a strong 1D demo, but the algorithm as printed is internally inconsistent and not reproducible until fixed.","tokens_in":18459,"tokens_out":4141,"would_cite":false,"duration_ms":43916,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65M60","68T07","35K55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sampling collocation points by local energy dissipation rate density, rather than by PDE residual, cuts the relative mean-square error of PINN solutions to the Allen-Cahn equation by up to a factor of six while using lower-order…","keywords":["physics-informed neural networks","energy dissipation rate","adaptive sampling","Allen-Cahn equation","dynamic boundary conditions","thermodynamically consistent models","phase field"],"falsifier":"Run EDRAS on a thermodynamically consistent Allen-Cahn problem with a slowly moving flat interface where the local dissipation rate is low but the interface-position error is large; if EDRAS fails to concentrate points there and matches RAR's error level, the guiding-metric assumption would be contradicted.","tokens_in":1470,"feed_emoji":"⚡","tokens_out":2220,"duration_ms":57555,"temperature":0.7,"pith_summary":"This paper introduces Energy Dissipation Rate guided Adaptive Sampling (EDRAS), a strategy for physics-informed neural networks that chooses new collocation points where the local energy dissipation rate density is largest instead of where the PDE residual is largest. The central claim is that for thermodynamically consistent models, regions of high energy dissipation mark where the approximate solution is far from the true solution, even when the residual there is small. On a one-dimensional Allen-Cahn test the method achieves up to a sixfold reduction in relative mean-square error compared with residual-based adaptive refinement, and it is computationally cheaper because it evaluates lower-order derivatives and avoids estimating a global residual distribution. The paper also uses EDRAS-enhanced PINNs to study how dynamic boundary conditions alter bulk phase evolution in disk- and ellipse-shaped domains, finding that higher surface mobility accelerates near-boundary bulk dynamics.","feed_headline":"Sampling by energy dissipation cuts PINN error sixfold","feed_subtitle":"Adaptive collocation points guided by local energy loss, not PDE residual, sharpen Allen-Cahn solutions in complex 2D domains.","key_machinery":"The load-bearing object is the local energy dissipation rate density $e_{edr}^f=\\phi_t^2/M_b$ for the bulk and $e_{edr}^b=\\phi_t^2/M_s$ for the boundary, used as an importance metric for resampling. The method computes it from the flux $\\phi_t$ rather than from the chemical potential $\\mu$, which requires higher-order derivatives, and it maintains domain coverage through a density-aware threshold that enforces a minimum number of points per subdomain while adding the highest-dissipation candidates. This machinery connects adaptive sampling to the Lyapunov structure of the free energy, concentrating training points where the solution's variance is large.","core_discovery":"The paper's central discovery is that the local energy dissipation rate density, computed as $\\phi_t^2/M_b$ in the bulk and $\\phi_t^2/M_s$ on the boundary, is a more reliable sampling guide than the PDE residual for PINNs solving thermodynamically consistent Allen-Cahn systems. In a probabilistic analysis, residual-based refinement with a limited budget selects only points with large residuals (groups B and D) and never the small-residual, large-error points (group A); EDRAS samples from both critical groups and attains a higher probability of selecting high-error points than either RAR or RAD. The numerical experiments show that EDRAS reduces the relative MSE from $6.06\\times10^{-4}$ to $1.08\\times10^{-4}$ in the 1D periodic case, and that even mixing EDRAS with RAR markedly improves accuracy over RAR alone.","pith_inferences":["An editorial extension is that the same dissipation-density heuristic should transfer to other gradient flows such as Cahn-Hilliard and phase-field crystal models, where $\\phi_t^2/M$ is still the natural local dissipation indicator.","If the dissipation-error correlation holds beyond Allen-Cahn, EDRAS could serve as a cheap a posteriori error estimator inside classical mesh adaptivity, not only inside neural-network training.","A testable extension is to compare EDRAS with RAD on a two- or three-dimensional thermodynamically consistent problem to verify that the computational saving grows with dimension, as the paper argues from the cost of the normalization factor.","A natural hybrid would weight dissipation density and residual together; the paper's group analysis suggests such a blend could hedge against regimes where a dynamically active region is already well approximated."],"forward_implications":["EDRAS can be attached to any PINN solver for a thermodynamically consistent model with a known free energy functional, without changing the network architecture or optimizer.","For the Allen-Cahn equation, EDRAS alone outperforms residual-based adaptive refinement on MSE, MAE, and $L^\\infty$ error, and combining it with RAR already gives most of the improvement.","Because EDRAS avoids estimating the normalization $\\int_\\Omega R(y)\\,dy$ needed by RAD and RAR-D, it is better suited to high-dimensional problems where that integral becomes intractable.","In 2D disk and ellipse domains, higher surface mobility in dynamic boundary conditions accelerates boundary coarsening and slightly accelerates bulk energy decay, indicating surface mobility can act as a control parameter for bulk dynamics.","The density threshold $d_f^{(0)}$ has a non-monotonic effect on accuracy: moderate increases reduce error, but too high a threshold degrades performance, so the method requires tuning this hyperparameter."],"supporting_citations":[{"why":"Supplies the residual-based adaptive refinement (RAR) method and the DeepXDE framework that serves as the main baseline for EDRAS.","marker":"[23]"},{"why":"Provides the systematic study of non-adaptive and residual-based adaptive sampling, including RAR-D and RAD, that EDRAS is compared against.","marker":"[36]"},{"why":"Introduces importance-sampling-based adaptive collocation point selection, a precursor distribution-based approach.","marker":"[27]"},{"why":"Derives thermodynamically consistent dynamic boundary conditions for phase field models via the generalized Onsager principle, giving the model EDRAS is applied to.","marker":"[17]"},{"why":"Provides the coupled bulk-surface thermodynamic modeling framework underlying the free energy and dissipation rate structure.","marker":"[16]"},{"why":"Supplies the finite-difference reference solutions used to measure EDRAS-enhanced PINN accuracy in 2D Neumann and dynamic boundary condition tests.","marker":"[39]"}],"fun_headline_variants":["Energy-loss sampling cuts PINN error sixfold","Dissipation-guided sampling beats residual PINN methods","Physical energy loss guides better PINN sampling","Dissipation metric sharpens Allen-Cahn PINN solutions"],"cache_read_input_tokens":20480,"weakest_assumption_plain":"The method works only if the places where the approximate neural-network solution dissipates energy fastest are also the places where its approximation error is largest; the paper demonstrates this correlation in one 1D example but does not prove it in general.","fun_headline_variants_meta":{"raw":{"variants":["Energy-loss sampling cuts PINN error sixfold","Dissipation-guided sampling beats residual PINN methods","Physical energy loss guides better PINN sampling","Dissipation metric sharpens Allen-Cahn PINN solutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000186,"raw_usage":{"total_tokens":1361,"prompt_tokens":1015,"completion_tokens":346,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":631,"completion_tokens_details":{"reasoning_tokens":286}},"tokens_in":631,"tokens_out":346,"duration_ms":4491,"temperature":1.0,"reasoning_tokens":286,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:48:13.627000+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run EDRAS on a thermodynamically consistent Allen-Cahn problem with a slowly moving flat interface where the local dissipation rate is low but the interface-position error is large; if EDRAS fails to concentrate points there and matches RAR's error level, the guiding-metric assumption would be contradicted.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the systematic study of non-adaptive and residual-based adaptive sampling, including RAR-D and RAD, that EDRAS is compared against."},{"cited_title":"Efficient training of physics-informed neural networks via importance sampling.Computer-Aided Civil and Infras- tructure Engineering, 36(8):962–977, 2021","cited_arxiv_id":null,"evidence_quote":"Introduces importance-sampling-based adaptive collocation point selection, a precursor distribution-based approach."},{"cited_title":"Thermodynamically consistent dynamic boundary conditions of phase field models.Communications in Mathematical Sciences, 21(3), 2023","cited_arxiv_id":null,"evidence_quote":"Derives thermodynamically consistent dynamic boundary conditions for phase field models via the generalized Onsager principle, giving the model EDRAS is applied to."},{"cited_title":"Thermodynamically consistent models for coupled bulk and surface dynamics.Entropy, 24(11):1683, 2022","cited_arxiv_id":null,"evidence_quote":"Provides the coupled bulk-surface thermodynamic modeling framework underlying the free energy and dissipation rate structure."},{"cited_title":"Variational embedding domain method for thermo- dynamically consistent allen-cahn model in arbitrary domains.submitted","cited_arxiv_id":null,"evidence_quote":"Supplies the finite-difference reference solutions used to measure EDRAS-enhanced PINN accuracy in 2D Neumann and dynamic boundary condition tests."}],"review_version":1}