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REVIEW 4 major objections 5 minor 63 references

Physics-informed neural networks for solving moving interface flow problems using the level set approach

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read PirateNet-based physics-informed neural networks solve the level set transport equation for Zalesak's disk and time-reversed vortex flow to relative $L^2$ errors of 0.14% and 0.85%, without upwind stabilization or mass-conservation schemes.

desk verdict Credible demonstration of PirateNet for level set benchmarks, but the headline L2 errors are measured against an unvalidated in-house FEM reference and partly in-sample, so the 'state-of-the-art' numbers are provisional. read the letter →

arxiv 2502.02440 v2 pith:MRLMUMFM submitted 2025-02-04 physics.comp-ph physics.flu-dyn

Mathieu Mullins , Hamza Kamil , Adil Fahsi , Azzeddine Soulaimani This is my paper · ORCID
classification physics.comp-phphysics.flu-dyn PACS 47.11.-j47.55.-t
keywords Physics-informedneuralnetworksLevelsetmethodMovinginterfaceflowsPirateNetZalesak'sdiskTime-reversedvortexflowSequence-to-sequencetrainingGeometricreinitialization
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 sets out to show that the level set transport equation $\partial \phi/\partial t + \mathbf{u}\cdot\nabla\phi=0$, the standard Eulerian tool for tracking moving interfaces, can be solved by a modern physics-informed neural network (PirateNet) even when the prescribed velocity field strongly stretches and deforms the interface. The authors report relative $L^2$ errors of $0.14\%$ for Zalesak's rotating slotted disk and $0.85\%$ for the time-reversed vortex flow, measured against a finite-element reference, with no upwind stabilization and no explicit mass-conservation term. They further find that Eikonal regularization and Monte-Carlo mass loss give little or no benefit on these benchmarks and can degrade the solution when weighted too heavily. For the harder coupled level set–Navier-Stokes dam-break problem, they propose a geometric reinitialization between sequence-to-sequence time windows to preserve the signed-distance property, obtaining $L^2=5.6\%$ at substantially higher computational cost than the finite-element reference. If these results hold, they make PINNs a plausible mesh-free alternative for interface tracking while also clarifying where neural solvers still trail classical methods.

What carries the argument

The machinery that carries the argument is the PirateNet architecture: a residual network whose adaptive skip connections are controlled by trainable parameters $\alpha^{(l)}$ that start at zero (each residual block is an identity map) and grow during training, so the effective depth of the network increases only when nonlinearity is needed. Around this core, the pipeline assembles random Fourier feature embeddings of $(x,t)$ coordinates, random weight factorization, physics-informed initialization (the final linear layer is fit by least squares to the initial condition), causal training (residual losses on later time chunks are weighted down until earlier chunks are learned), gradient-normalization loss balancing, and sequence-to-sequence training that divides $[0,T]$ into windows, feeding each window's initial condition from the previous window's prediction. The level set function $\phi(x,t)$ is a signed distance function whose zero contour is the interface, transported by the advection equation; the Eikonal property $\|\nabla\phi\|=1$ appears only as an optional regularization term. For the dam-break case, a geometric reinitialization using the Euclidean distance transform restores the signed-distance property between windows while leaving the zero level set in place.

What would settle it

Run the trained 'Sota' model on the time-reversed vortex flow and compare its prediction at $t=T$, when the flow has reversed and the initial circle should be recovered exactly, against the known circular interface: if the returned interface deviates from the circle by far more than the reported $0.85\%$ $L^2$ error, or if the same error persists when measured against an independent high-order reference instead of the paper's finite-element solution, the central claim is falsified.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the level set advection equation $\partial \phi/\partial t + \mathbf{u}\cdot\nabla\phi = 0$ can be learned end-to-end by a PirateNet-based PINN even when the velocity field is strongly vortical and the interface is heavily stretched, without the upwind stabilization or reinitialization that classical discretizations typically require. The error tables carry the argument: for Zalesak's disk, PirateNet gives $L^2=0.35\%$ versus $2.96\%$ for the improved PINN and $4.18\%$ for the original PINN, and a hyperparameter-tuned 'Sota' configuration reaches $0.14\%$; for the time-reversed vortex flow, the same ordering gives $5.24\%$, $20.86\%$, and $51.20\%$, with Sota at $0.85\%$. The authors interpret this as the first PINN-based level set solution in a complex, varying, vortical velocity field. They also show that adding an Eikonal loss ($\|\nabla\phi\|=1$) or a Monte-Carlo mass-loss term does not improve accuracy on these benchmarks and can freeze the interface or fail to converge when the weight is too large. For the coupled level set–Navier-Stokes dam-break problem, where the signed-distance property decays during training, they embed a geometric reinitialization step between sequence-to-sequence windows—resetting the level set to its signed distance while keeping the zero contour fixed—and report $L^2=5.6\%$ with no growing error pattern.

Load-bearing premise

The reported error numbers are computed against the authors' own finite-element reference solution, so the central claim assumes that reference is accurate enough to serve as ground truth; if it is not, the low $L^2$ values and the conclusion that no upwind stabilization or mass conservation is needed do not follow.

Editorial extensions

If this is right

  • If the reported errors are representative, level-set advection in strongly deforming flows can be solved by PINNs without upwind stabilization, so the classical artificial-diffusion burden of transport schemes does not transfer to this neural formulation.
  • Architecture and training recipe matter: on both benchmarks the PirateNet configuration reduces the relative $L^2$ error by roughly an order of magnitude compared with the original PINN, with the widest gap on the strongly stretching vortex flow ($0.85\%$ versus $51.20\%$).
  • The Eikonal and mass-loss regularizations tested here are not needed for the conservative benchmarks and can degrade accuracy or fail to converge, so the default recommendation is to omit them unless a specific case requires them.
  • For the coupled level set–Navier-Stokes dam-break problem, embedding a geometric reinitialization between sequence-to-sequence windows gives stable long-time inference at $L^2=5.6\%$, but at substantially higher wall-clock cost than the finite-element reference (16 hours on a V100 GPU versus 1.25 hours on a laptop CPU).
  • The paper's claim is about interface-position accuracy, not mass conservation: the PINN's mean absolute percent mass error is $1.18\%$ on the vortex test versus $0.07\%$ for the finite-element solver.

Reading between the lines

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

  • A natural next check, not run in the paper, would compare the tuned network on the time-reversed vortex against the exact final state (the flow returns the circle to its initial position), separating architecture performance from the accuracy of the finite-element reference.
  • The geometric reinitialization turns the coupled solver into a hybrid: the PINN no longer solves the full space-time problem in one shot, so the mesh-free advantage is partially traded against a discrete reset; whether a learned reinitialization could replace the Euclidean distance transform remains untested.
  • The observed tendency of a strong Eikonal weight to freeze the interface suggests that a dynamic schedule—Eikonal weight high at early times and relaxed once the interface starts to deform—might recover both signed-distance quality and stretching, an ablation the paper does not report.
  • Because mass conservation is the main quantitative gap, the most promising next step is a differentiable mass-loss term evaluated during training rather than the separately trained second network, which the paper observes fails to converge; such a formulation could close the gap without the overhead the paper measured.
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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 paper proposes a physics-informed neural network framework based on the PirateNet architecture, extended with causal training, sequence-to-sequence learning, random weight factorization, and Fourier feature embeddings, for solving level set transport equations in moving interface problems. The method is tested on Zalesak's rotating disk and the time-reversed vortex flow, with reported relative L2 errors of 0.14% and 0.85% against a SUPG-stabilized finite element reference, and on a coupled level set–Navier-Stokes dam break problem with a proposed geometric reinitialization step. The central claim is that PirateNet-based PINNs can solve level set problems with complex interface deformation without upwind stabilization or explicit mass conservation, and that the reported errors are state-of-the-art.

Significance. If the quantitative claims hold, this is a useful contribution: it shows that modern PINN training techniques can handle the transport of sharp interfaces in highly deforming flows, and it provides a careful ablation of Eikonal and mass-loss regularizers, concluding that they do not help in most tested cases. The paper also demonstrates a practical geometric reinitialization strategy for long-time coupled simulations. The main empirical finding that PirateNet outperforms plain and default PINNs is supported by the reported experiments, and the honesty in reporting negative results for mass and Eikonal losses is a strength. However, the state-of-the-art error numbers are not yet anchored because the reference solution is an unvalidated in-house FEM result in cases where exact solutions exist, and the headline comparison is confounded by asymmetric hyperparameter tuning. Reproducibility is limited by the statement that code is available only upon request.

major comments (4)
  1. [Secs. 4.1.2, 4.2, and Eq. (40); Tables 1 and 3] The headline L2 errors are computed against a SUPG/T6 FEM reference generated by the authors (Fahsi and Soulaimani, 2017) on 10,470 elements, with no mesh-convergence study and no independent validation of that reference. This is load-bearing because for both benchmarks the exact final state is known analytically: the velocity (41) is a rigid rotation of the initial condition, and the stream function (43) makes phi(x,T) = phi(x,0). The paper should report the L2 error of both the PINN and the FEM reference relative to these exact states, or validate the FEM reference against an independent high-resolution solver, before claiming 0.14% and 0.85% as state-of-the-art. Without this, the low error numbers may partly reflect agreement with a diffused or phase-shifted reference rather than the true solution.
  2. [Sec. 4, Table B.11, Table B.12, Tables 1 and 3] The 'Sota' configuration is selected by a Bayesian hyperparameter sweep (WandB) on the same test benchmark whose error is then reported, while 'Plain', 'Default', and 'PirateNet' use fixed default hyperparameters without an equivalent sweep. In addition, Table B.11 shows that Sota for Zalesak's disk trains for 80,000 steps versus 20,000 for the other configurations. This asymmetric tuning means the comparisons in Tables 1 and 3 do not isolate the architecture's contribution and make the reported Sota error partly in-sample. Please either apply the same sweep to all configurations, or clearly label the Sota results as test-set-selected and provide a like-for-like fixed-hyperparameter comparison.
  3. [Tables 1, 3, A.6, A.7] All reported errors come from single runs without multiple seeds or error bars. PINN training is initialization- and optimizer-dependent, so claims such as 'PirateNet performs better than Default' and the 2.49% vs 2.52% difference in Table A.8 cannot be assessed for statistical significance. At least a small number of seeds with mean plus/minus standard deviation should be reported for the main comparisons.
  4. [Sec. 4.3.4 and Sec. 4.3.1] The dam break error statement says 'the frames considered in this error calculation are only the ones subjected to the reinitialization step,' which is ambiguous and potentially selective; if error is evaluated only on a subset of frames chosen after the fact, the reported L2 = 5.6% is not a representative full-time error. Please specify exactly which frames are included and also report the error over all time frames. Further, the momentum residuals (56)-(57) drop all viscous terms involving derivatives of mu, with the justification that residuals are not computed exactly on the interface; a quantitative argument for why these terms are negligible in the loss for this two-phase problem would strengthen the methodology.
minor comments (5)
  1. [Sec. 4.1.1] The text states that 'no enhancements to PINN training, as discussed in Section 3.2, were applied in this example,' but the same paragraph adopts a sequence-to-sequence approach, which is introduced as an enhancement in Section 3.2.5; please clarify what 'no enhancements' means in this context.
  2. [Code availability] The code is described as 'available upon request'; for a computational journal paper, providing a permanent repository link would substantially improve reproducibility.
  3. [Sec. 3.3.2, Eq. (35)] The Monte Carlo area estimator uses a uniform random sample, but no variance reduction or number of points N is specified in the main text; please state the sample sizes used for the mass loss computations in Sections 4.1.2 and 4.2.
  4. [Throughout] There are several typographical issues and formatting inconsistencies, for example 'PirateNets'' in the introduction, 'W andB’s' in the text, and inconsistent use of 'PirateNet' versus 'PirateNets'; a careful copyedit is recommended.
  5. [Sec. 4.1.2, Figure 5] The mass error comparison between Sota and FEM is informative, but the figure would be clearer if the two curves were on a common axis with confidence intervals or repeated runs, given the stated high variance of the PINN mass error.

Circularity Check

2 steps flagged · score 4.0 of 10

Headline L2 errors are partly in-sample (Sota hyperparameters selected on the same benchmarks) and are measured against an unvalidated, self-cited FEM reference, though the core PINN derivation itself is not circular.

  1. fitted input called prediction [Sec. 4.1.2 (Zalesak's disk), Sec. 4.2 (time-reversed vortex), Tables 1 and 3, Tables B.11 and B.12]
    "With the PirateNet architecture, we then performed a Bayesian hyperparameter sweep using Wandb's sweep module [59]. Table B.11's Sota configuration summarizes the optimal hyperparameters, allowing us to lower the error down to L2 Sota = 0.14%."

    The 'Sota' configuration is selected by minimizing the same relative L2 error of Eq. (40) on the very benchmark whose error is then reported as the headline result. The same procedure is repeated in Sec. 4.2: 'With the PirateNet architecture, we then performed a Bayesian hyperparameter sweep ... With Sota, we were able to reduce the error down to L2 Sota = 0.85%.' Thus the reported accuracy is an in-sample selection score: the hyperparameters are effectively fitted to the reported quantity, so the low error is partly forced by the selection procedure rather than being an independent prediction.

  2. self citation load bearing [Sec. 4.1.2, Sec. 4.2, Sec. 4.3.4, Eq. (40), Refs. [45] and [10]]
    "The reference solution was obtained using the SUPG-stabilized finite element method, with a quadratic approximation for the level set function on unstructured triangular elements (T6) and a third-order explicit strong–stability–preserving Runge–Kutta (SSP–RK) time integration scheme [45] implemented with MATLAB."

    Every headline error is defined against this reference via Eq. (40): 'Relative L2 Error = ‖uPINN − ureference‖2 / ‖ureference‖2.' The reference solver is [45] (Fahsi and Soulaimani), i.e., the same research group's FEM code; the contribution statement confirms 'Adil Fahsi: FEM code, validation, review.' For both benchmarks exact terminal states are available (rigid rotation for Zalesak's disk; time-reversed stream function gives φ(x,T)=φ(x,0) for the vortex), but the paper never anchors the reference to these states. Consequently the 'state-of-the-art' numbers are deviations from an unvalidated, self-cited simulation rather than from an externally established solution, making the central accuracy claim load-bearing on a self-citation.

full rationale

The paper's mathematical core is not circular. The PINN loss in Eqs. (29)-(30) is the level set PDE residual plus the initial condition; no reference data enter the loss, and the solution is produced by minimizing that residual. The Burgers' validation in Appendix A uses external references (Chebfun, finite differences) and shows the same qualitative architecture comparisons. The two flagged issues affect the quantitative 'state-of-the-art' claim, not the derivation itself. First, the Sota hyperparameters are selected by a Bayesian sweep that minimizes the same Eq. (40) error on the same benchmark later reported, so the 0.14% and 0.85% figures are in-sample selection scores rather than out-of-sample predictions. Second, the reference solution used in Eq. (40) for both level set benchmarks is generated by the authors' own FEM solver (Ref. [45], with co-author Adil Fahsi listed as the FEM-code contributor), and the paper does not validate that reference against the known exact terminal states. These are genuine independence problems for the headline error numbers, but they do not make the PDE-solving procedure equivalent to its inputs by definition. Score 4 reflects partial circularity in the accuracy claim while acknowledging that the central methodological content retains independent substance.

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

The central claims rest on a small set of tuned hyperparameters, the accuracy of a self-generated FEM reference, and standard level set assumptions. No new physical entities are introduced.

free parameters (5)
  • Sota hyperparameters for Zalesak's disk = 8 residual blocks, Swish, Fourier scale 1.0, causal tolerance 1.0, RWF (1.0, 0.1), 80k steps
    Selected by WandB Bayesian sweep on the Zalesak benchmark itself; the reported L2=0.14% is the best of the sweep, not an out-of-sample error.
  • Sota hyperparameters for time-reversed vortex flow = 8 residual blocks, Swish, Fourier scale 2.0, causal tolerance 1.5, RWF (1.0, 0.1), 20k steps per window
    Selected by WandB Bayesian sweep on the vortex benchmark; L2=0.85% is the best of the sweep.
  • Eikonal regularization weight = scanned 1e-4 to 1.0
    The paper tests many weights and concludes the weight must be carefully chosen; high weights hurt the solution, and no improvement is seen for the benchmarks.
  • Mass loss weight = Method 1 with 0.1 to 100, Method 2 with 1000 iterations
    Explored but does not improve the benchmark errors; included to document the negative result.
  • S2S windows and reinitialization schedule for dam break = 16 windows, reinitialization between windows, 20k steps per window (40k for first)
    Chosen by hand; no ablation shows the sensitivity of the dam break L2=5.6% to this schedule.
assumptions (5)
  • standard math The level set transport equation (Eq. 6) with the given incompressible velocity fields fully describes the interface evolution for the benchmarks.
    This is the governing PDE from Osher and Sethian; the paper solves it with PINN residuals.
  • domain assumption The authors' own SUPG-stabilized FEM solver (ref [45]) produces reference solutions accurate enough to serve as ground truth for the relative L2 error.
    Invoked in Secs. 4.1.2, 4.2, and 4.3.4 via Eq. (40); the FEM solver is not independently validated in this paper.
  • domain assumption The signed distance property (Eikonal) is a useful target and can be restored by geometric reinitialization without moving the zero contour.
    Standard in level set literature; used in the proposed geometric reinitialization for the dam break.
  • ad hoc to paper In the coupled dam break equations, derivatives of viscosity are dropped because residuals are not computed exactly on the sharp interface (Sec. 4.3.1).
    The authors state this assumption explicitly to simplify the momentum residuals; its validity is not tested.
  • domain assumption The neural network can represent the level set solution with sufficient accuracy over the whole spatial domain and time windows; optimization via Adam or SOAP reaches a good local minimum.
    Implicit in all PINN training; no convergence guarantee.

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Cite this review

Pith. "Pith review of Physics-informed neural networks for solving moving interface flow problems using the level set approach." pith.science (2026). https://pith.science/paper/MRLMUMFM

@misc{pith2026250202440,
  author       = {Pith},
  title        = {Pith review of: Physics-informed neural networks for solving moving interface flow problems using the level set approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MRLMUMFM}},
  note         = {Machine review of arXiv:2502.02440}
}
abstract

This paper advances the use of physics-informed neural networks (PINNs) architectures to address moving interface problems via the level set method. Originally developed for other PDE-based problems, we particularly leverage PirateNet's features, including causal training, sequence-to-sequence learning, random weight factorization, and Fourier feature embeddings, and tailor them to handle problems with complex interface dynamics. Numerical experiments validate this framework on benchmark problems such as Zalesak's disk rotation and time-reversed vortex flow. We demonstrate that PINNs can efficiently solve level set problems exhibiting significant interface deformation without the need for upwind numerical stabilization, as generally required by classic discretization methods, or additional mass conservation schemes. However, incorporating an Eikonal regularization term in the loss function with an appropriate weight can further enhance results in specific scenarios. Our results indicate that PINNs with the PirateNet architecture surpass conventional PINNs in accuracy, achieving state-of-the-art error rates of $L^2=0.14\%$ for Zalesak's disk and $L^2=0.85 \%$ for the time-reversed vortex flow problem, as compared to reference solutions. Additionally, for a complex two-phase flow dam break problem coupling the level set with the Navier-Stokes equations, we propose a geometric reinitialization method embedded within the sequence-to-sequence training scheme to ensure long-term stability and accurate inference of the level set field.

Figures

Figures reproduced from arXiv: 2502.02440 by the authors.

Figure 1
Figure 1. Illustration of a two-phase domain (a), with Ω [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. PirateNets’ architecture [39]. The input coordinates are passed to a coordinate embedding before going through two dense layers and the residual block. The residual block is contained within the grey area in the figure and is repeated L times. The initial dense layers are passed to two gating operations in orange within the residual block. The adaptive skip connection is affected by the trainable parameter α and con… view at source ↗
Figure 3
Figure 3. Visualization of the PINN results: (Top) Profiles showing the impact of Eikonal regularization. (Middle) [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: Zalesak’s disk reference and Sota solution evolution (T = 2π s) [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Zalesak’s disk absolute percent mass error evolution ( [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Zalesak’s disk: influence of the Eikonal loss term [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Level set vortex: Reference, predicted, and absolute error of [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: Level set vortex: Evolution of mass loss with fully trained [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]
Figure 9
Figure 9. Figure 9: Level set vortex: influence of the Eikonal loss term [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: Level set vortex: Test Eikonal loss term. [PITH_FULL_IMAGE:figures/full_fig_p023_10.png]
Figure 11
Figure 11. Figure 11: Level set vortex: Evolution of mass loss during training (2 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png]
Figure 12
Figure 12. Figure 12: Coupled level set-NS: Temporal evolution of [PITH_FULL_IMAGE:figures/full_fig_p028_12.png]
Figure 13
Figure 13. Figure 13: Coupled level set-NS: Temporal evolution of mass loss [PITH_FULL_IMAGE:figures/full_fig_p029_13.png]

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