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REVIEW 4 major objections 5 minor 41 references

Energy Dissipation Rate Guided Adaptive Sampling for Physics-Informed Neural Networks: Resolving Surface-Bulk Dynamics in Allen-Cahn Systems

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict A genuinely new PINN sampling idea with a strong 1D demo, but the algorithm as printed is internally inconsistent and not reproducible until fixed. read the letter →

arxiv 2507.09757 v1 pith:Y3MU7BH6 submitted 2025-07-13 math.NA cs.LGcs.NA

Chunyan Li , Wenkai Yu , Qi Wang This is my paper · ORCID
classification math.NAcs.LGcs.NA MSC 65M6068T0735K55
keywords physics-informedneuralnetworksenergydissipationrateadaptivesamplingAllen-Cahnequationdynamicboundaryconditionsthermodynamicallyconsistentmodelsphasefield
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [Algorithm 1, steps 2-3 and 8; Eq. (3); Eq. (11); Section 2.2] 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.
  2. [Section 3.2, Table 2; Section 3.3; Abstract] 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.
  3. [Section 3.3, Figures 10-13; Section 3.2] 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.
  4. [Algorithm 1; Section 2.2; Section 3.2, Table 3] 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.
minor comments (5)
  1. [Figure 7 caption, Section 3.1] 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.
  2. [Section 3.1, probability discussion] 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.
  3. [Eq. (3) and Eq. (18)] 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.
  4. [Algorithm 1, step 3] The phrase 'density of bulk energy dissipation rate density' is redundant; consider using 'bulk energy dissipation-rate density' throughout the algorithm.
  5. [Data Availability] 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.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing circularity: EDRAS accuracy is benchmarked against an external FDM reference; self-citations are non-load-bearing; Algorithm 1's eedrdf = phi_t/M_b is inconsistent with the dissipation density phi_t^2/M_b of Eq. (3), a reproducibility flaw rather than a circular step.

full rationale

The paper's central numerical claims — the up-to-sixfold relative-MSE reduction of EDRAS over RAR (Table 1) and the higher sampling probability that EDRAS lands in the high-error groups A union B (Figure 7) — are evaluated against an external finite-difference reference, not against the method's own training data: the paper states that the reference solution is computed using the finite difference method (Section 3.1), and the 2D validation similarly benchmarks against a reference solution obtained using finite difference methods [39]. The dissipation-to-error correspondence that motivates EDRAS is presented as an empirically tested heuristic (Figure 2; the group-probability experiment with error threshold e0 = 0.001 and residual threshold R0 equal to the mean residual), not derived as an identity, so sampling by dissipation density is not equivalent by construction to reducing the FDM-based error. The self-citations are real but not load-bearing: the governing system (2) and the dissipation identities (3), (11), (18), (20) are re-derived in-paper by direct calculation, so the thermodynamic-consistency premise does not rest solely on [3] (first author's dissertation) or [16,17] (Jing and Wang, with the corresponding author); and the cited well-posedness results are external ([2,7,8,9,29,30,37]). Reference [39] is a submitted same-group paper, but finite differences are an independent discretization paradigm, externally falsifiable, hence genuine evidence under the stated rules. Two caveats, neither an input-output reduction: (i) Algorithm 1, step 3, defines the sampling metric as eedrdf = phi_t/M_b, whereas the text defines it as the absolute value of the energy dissipation rate density in the bulk, which Eq. (3) gives as phi_t^2/M_b; because phi_t changes sign under Allen-Cahn dynamics, the printed algorithm is ambiguous (phi_t/M_b vs |phi_t|/M_b vs phi_t^2/M_b), and the choice changes which points are pruned and added, clouding reproducibility of Table 1 and Figures 5 and 7 — a correctness flaw, not circularity. (ii) The dynamic-boundary-condition insights (Section 3.3, Figures 10-13) are read off PINN+EDRAS solutions without an independent dynamic-BC solver, leaving them unvalidated rather than self-referential. No derivation step collapses into its own input, so the circularity score is minimal.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central numerical claim depends on the correlation between dissipation rate density and error (Axiom 2) and on several tunable parameters (1/3 threshold, density thresholds, subdomain count). The model itself is taken from prior work. No new physical entities are introduced.

free parameters (4)
  • Dissipation threshold fraction 1/3 = 1/3
    In Algorithm 1 step 2, the threshold is set to one third of the mean dissipation rate density over the training set. This factor is ad hoc; no sensitivity analysis is provided, and the error results may depend on it.
  • Density threshold d0_f (and d0_b) = 100-300 in experiments (Table 3)
    Minimum number of collocation points per subdomain. Table 3 shows a non-monotonic effect on error, so this is an important tunable hyperparameter without a clear selection rule.
  • Number of subdomains N for density detection = not specified
    The domain is divided into N subdomains to check sampling density; N is not given, yet it affects the sampling distribution and computational cost.
  • Time-marching segment breakpoints = T1=0.01, T2-T6 in [0.2,1.0] in 1D; uniform dt=0.2 with initial 0.01 in 2D
    The scheduling of temporal segments is a hyperparameter that can change accuracy; no systematic study is presented.
assumptions (5)
  • domain assumption Generalized Onsager principle yields thermodynamically consistent Allen-Cahn equations with dynamic boundary conditions (system 2).
    This is taken from the authors' earlier work [16,17] and is not re-derived here; the paper relies on it to define both the model and the energy dissipation rate that anchors EDRAS.
  • ad hoc to paper The local energy dissipation rate density of the approximate PINN solution is a reliable proxy for the local approximation error.
    Motivated by Figure 2 on a single 1D example; no proof or counterexample analysis is given. This is the core justification for EDRAS.
  • domain assumption The finite difference reference solutions are accurate enough to serve as ground truth.
    Used in 1D and in 2D for the Neumann case; the 2D FDM code is from an unpublished submitted manuscript [39] by the same group, so the reference itself is not independently verifiable.
  • standard math The frequency principle / spectral bias of neural networks describes the training difficulty (low to high frequencies).
    Invoked in the introduction to motivate why high-frequency structures are hard to learn; standard in the deep learning literature [38].
  • standard math The Allen-Cahn equation with double-well potential is thermodynamically consistent and has an energy dissipation inequality (equation 3).
    Standard in phase-field modeling; the paper uses this to define the dissipation rate metric.

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Cite this review

Pith. "Pith review of Energy Dissipation Rate Guided Adaptive Sampling for Physics-Informed Neural Networks: Resolving Surface-Bulk Dynamics in Allen-Cahn Systems." pith.science (2026). https://pith.science/paper/Y3MU7BH6

@misc{pith2026250709757,
  author       = {Pith},
  title        = {Pith review of: Energy Dissipation Rate Guided Adaptive Sampling for Physics-Informed Neural Networks: Resolving Surface-Bulk Dynamics in Allen-Cahn Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y3MU7BH6}},
  note         = {Machine review of arXiv:2507.09757}
}
read the original abstract

We introduce the Energy Dissipation Rate guided Adaptive Sampling (EDRAS) strategy, a novel method that substantially enhances the performance of Physics-Informed Neural Networks (PINNs) in solving thermodynamically consistent partial differential equations (PDEs) over arbitrary domains. EDRAS leverages the local energy dissipation rate density as a guiding metric to identify and adaptively re-sample critical collocation points from both the interior and boundary of the computational domain. This dynamical sampling approach improves the accuracy of residual-based PINNs by aligning the training process with the underlying physical structure of the system. In this study, we demonstrate the effectiveness of EDRAS using the Allen-Cahn phase field model in irregular geometries, achieving up to a sixfold reduction in the relative mean square error compared to traditional residual-based adaptive refinement (RAR) methods. Moreover, we compare EDRAS with other residual-based adaptive sampling approaches and show that EDRAS is not only computationally more efficient but also more likely to identify high-impact collocation points. Through numerical solutions of the Allen-Cahn equation with both static (Neumann) and dynamic boundary conditions in 2D disk- and ellipse-shaped domains solved using PINN coupled with EDRAS, we gain significant insights into how dynamic boundary conditions influence bulk phase evolution and thermodynamic behavior. The proposed approach offers an effective, physically informed enhancement to PINN frameworks for solving thermodynamically consistent models, making PINN a robust and versatile computational tool for investigating complex thermodynamic processes in arbitrary geometries.

Figures

Figures reproduced from arXiv: 2507.09757 by the authors.

Figure 1
Figure 1. Left: Numerical solutions of the Allen-Cahn equation at [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Top Left: The numerical solution obtained from FDM. Top Right: The energy dissipation [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Top: contour plots of solutions obtained using PINNs with RAR (left), EDRAS (middle), [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (19 more)
Figure 4
Figure 4. Figure 4: Snapshots of solutions obtained from PINN with RAR (top), EDRAS (middle) and both [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Top: 3 examples of re-sampled collocation points at [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Schematics of the RAR (left), RAD (middle), and EDRAS (right) method, respectively. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: The sampling probabilities of the 4 groups, [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]
Figure 8
Figure 8. Figure 8: Comparison of FDM and EDRAS-enhanced PINN solutions of the Allen-Cahn equation [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Snapshots of solutions at selected time with different mobility values to showcase the [PITH_FULL_IMAGE:figures/full_fig_p020_9.png]
Figure 10
Figure 10. Figure 10: Plots of the total energy (blue), bulk energy (red), and surface energy (purple) with [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: Time evolves of total energy, bulk energy, and surface energy for Allen-Cahn equation with [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]
Figure 12
Figure 12. Figure 12: Snapshots of solutions at selected time at [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Snapshots of solutions at selected time at [PITH_FULL_IMAGE:figures/full_fig_p024_13.png]
Figure 14
Figure 14. Figure 14: Snapshots of solutions obtained using PINN with RAR (top), EDRAS (middle) and [PITH_FULL_IMAGE:figures/full_fig_p028_14.png]
Figure 15
Figure 15. Figure 15: Snapshots of solutions obtained using PINN with RAR (top), EDRAS (middle) and [PITH_FULL_IMAGE:figures/full_fig_p029_15.png]
Figure 16
Figure 16. Figure 16: Snapshots of solutions obtained using PINN with RAR (top), EDRAS (middle) and [PITH_FULL_IMAGE:figures/full_fig_p029_16.png]
Figure 17
Figure 17. Figure 17: Snapshots of solutions obtained using PINN with RAR (top), EDRAS (middle) and [PITH_FULL_IMAGE:figures/full_fig_p030_17.png]
Figure 18
Figure 18. Figure 18: Probabilities of the 4 groups (A, B, C, D) for RAR, RAD and EDRAS in domain [0, 0.1]× [−1, 1] plotted over training epochs. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_18.png]
Figure 19
Figure 19. Figure 19: Probabilities of the 4 groups for RAR, RAD and EDRAS in domain [PITH_FULL_IMAGE:figures/full_fig_p031_19.png]
Figure 20
Figure 20. Figure 20: Probabilities of the 4 groups for RAR, RAD and EDRAS in domain [PITH_FULL_IMAGE:figures/full_fig_p031_20.png]
Figure 21
Figure 21. Figure 21: Probabilities of the 4 groups for RAR, RAD and EDRAS in domain [PITH_FULL_IMAGE:figures/full_fig_p031_21.png]
Figure 22
Figure 22. Figure 22: Probabilities of the 4 groups for RAR, RAD and EDRAS in domain [PITH_FULL_IMAGE:figures/full_fig_p032_22.png]

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