{"id":"f794ff2f-2146-4749-a9e6-54994203f377","arxiv_id":"2607.27349","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A PINN framework solves 2D plane-wave scattering for both TM and TE polarizations, with a tanh-smoothed permittivity used to recover TE accuracy at dielectric boundaries.","lead":"This paper trains physics-informed neural networks to solve 2D light-scattering problems in arbitrary dielectric structures, reporting small errors for TM polarization and improved TE accuracy after smoothing material boundaries. A generalist should read it as a test of whether meshless neural solvers can replace conventional meshed electromagnetic simulators.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The printed boundary residual Eq. (26) enforces the wrong radiation condition on the x_min and y_min edges: with sigma = -1 and the outward normal it gives d_x u = +i k0 u instead of the stated d_x u = -i k0 u, so the reported accuracies cannot be reproduced from the methods as written.","rationale":"The reader's weakest assumption is correct and is the most load-bearing issue. I re-derived Eq. (26) against Eqs. (22)-(23): at x_min the outward normal is -d_x, and sigma = -1, so the residual vanishes for d_x u = +i k0 u, the opposite of Eq. (23); the same reversal occurs at y_min. This is not an optional sign convention, because the time dependence is fixed as e^{-i omega t}, which fixes the signs in Eqs. (22)-(23). A literal implementation would therefore inject waves propagating into the domain at the left and bottom boundaries. Since no code or data are included, the reported TM errors (0.07-0.15) cannot be checked against the printed algorithm. I also checked the TM/TE volume residuals (Eqs. (10)-(18)): they are internally consistent, and the tanh smoothing idea is physically motivated, so the paper has salvageable content. However, the central algorithm description is inconsistent, and the TE benchmarks are weak (Table I: relative L2 errors 0.92 and 1.09), with no quantitative arbitrary-structure TE error even after smoothing. These considerations support the reader's REJECT; the verdict remains REJECT, so no adjustment is needed.","tokens_in":14753,"tokens_out":17829,"duration_ms":153979,"concrete_test":"Use the exact plane wave u_sc = e^{-i k0 (x - x_min)} on the left edge, which identically satisfies Eq. (23)'s condition (d_x + i k0) u_sc = 0. Insert it, with sigma = -1 and d_n = -d_x, into Eq. (26). The residual evaluates to (2 sin(k0 (x - x_min)), 2 cos(k0 (x - x_min))), which is nonzero, proving that the printed boundary residual does not implement the stated radiation condition. Repeating at y = y_min with e^{-i k0 (y - y_min)} gives the same conclusion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing defect is the printed boundary residual. The stated first-order radiation condition at x = x_min is (d_x + i k0) u_sc = 0 (Eq. (23)), which for u_sc = u^Re + i u^Im is equivalent to d_x u^Re - k0 u^Im = 0 and d_x u^Im + k0 u^Re = 0. At x = x_min the outward normal derivative is d_n = -d_x. Substituting this and sigma = -1 into Eq. (26) gives R_BC^Re = -(d_x u^Re / k0 + u^Im) and R_BC^Im = -(d_x u^Im / k0 - u^Re). Setting both to zero yields d_x u^Re + k0 u^Im = 0 and d_x u^Im - k0 u^Re = 0, i.e. d_x u = +i k0 u. This is the opposite of Eq. (23), which requires d_x u = -i k0 u at the left edge; it drives the scattered field to a rightward (inward-propagating) wave at x_min and y_min, not to the outgoing radiation condition. The residual would be correct with sigma = +1 on all four edges when d_n is the outward normal, so the sigma prescription in the text is internally inconsistent. Because Eq. (26) is the only algorithmic statement of the boundary implementation and no code or data are provided, the numerical results cannot be checked against the method as written; a literal implementation should produce reflected or inward waves at the left and bottom boundaries, which is incompatible with the reported TM relative L2 errors of 0.07-0.15.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a meshless physics-informed neural network (PINN) framework for two-dimensional time-harmonic electromagnetic wave scattering in inhomogeneous dielectric media. The network is trained by minimizing residuals of the frequency-domain scattered-field equations together with a first-order radiation boundary condition, without using any target field data. The authors derive separate scattered-field formulations for TM (Ez) and TE (Hz) polarizations, introduce a hyperbolic-tangent smoothing of the permittivity to treat TE interface singularities, and validate the method against analytical solutions for a single cylinder, concentric core-shell cylinders, and three-cylinder configurations, and against FDTD for an irregular composite structure. Reported TM relative L2 errors are mostly below or about 0.1, while unsmoothed TE errors are large (0.20-1.09); the paper claims that permittivity smoothing restores TE accuracy, with qualitative FDTD agreement shown for the irregular case.","tokens_in":15137,"tokens_out":12079,"duration_ms":89375,"significance":"If the technical issues are resolved, the paper would be a useful contribution to the PINN-for-electromagnetics literature. The derivation of the scattered-field PDEs for both polarizations is clear and the TM benchmark results are encouraging. The paper commits to a strictly physics-residual-based training procedure and provides comparisons against independent analytical and FDTD benchmarks. However, the printed boundary residual in Eq. (26) contains a sign error that reverses the radiation condition on two edges, so the method as described cannot reproduce the reported accuracies. In addition, the central claim for the TE polarization rests on visual comparisons rather than quantitative error metrics, and the smoothing-width selection is benchmark-dependent. With corrected equations and quantitative TE results, the contribution would be solid; in its current form, the numerical claims are not fully verifiable from the manuscript.","major_comments":[{"comment":"The boundary residual defined in Eq. (26) is inconsistent with the stated first-order radiation condition (23) on the x=x_min and y=y_min edges. With sigma=-1 and the outward normal (for example, d/dn = -d/dx at x=x_min), setting the residual to zero yields d_x u^Re = -k0 u^Im and d_x u^Im = +k0 u^Re, which is equivalent to d_x u = +i k0 u. This is the opposite of the condition (d_x + i k0)u = 0 stated in Eq. (23), which requires d_x u = -i k0 u. The same sign inversion occurs at y=y_min. As written, the loss would drive the scattered field toward an inward-propagating wave at the left and bottom edges, so the reported L2 errors against the analytical and FDTD benchmarks cannot be reproduced from the printed method. The correct implementation would use sigma=+1 on all four edges when d/dn is the outward normal (or an equivalent sign convention). The authors must correct Eq. (26) and clearly specify the residual on each edge.","section":"Sec. III, Eq. (26)"},{"comment":"The paper's central claim for the TE polarization, namely that tanh smoothing removes the interface singularities and yields results \"closely matching\" the FDTD method, is not supported by any quantitative error metric. Table I reports unsmoothed TE errors of 0.9184, 0.1983, and 1.0936 for the three benchmark cases; these values are consistent with a failed fit rather than a working model. After introducing the smoothing function, the paper shows only qualitative field maps (Fig. 7) for the arbitrary-scatterer case and states that a particular delta 'minimizes epsilon_L2', but it never reports the numerical epsilon_L2 value for the smoothed TE solutions. Without this number, or equivalent values for the smoothed cylinder cases, the claimed improvement cannot be evaluated.","section":"Sec. IV.B and Table I"},{"comment":"The transition width delta is effectively selected by minimizing the relative L2 error against the benchmark solution (Fig. 5), and the text describing Fig. 7 states that delta=10^-3 was chosen as the value that minimizes epsilon_L2. This makes the subsequent FDTD comparison a form of calibration rather than a predictive test. The authors should clarify whether delta is a free parameter tuned on each benchmark case and, if so, discuss the implications for the reported error levels. Ideally, the smoothing width should be determined by a physics-based criterion (e.g., resolving the interface on a scale consistent with the discretization) or evaluated on a separate validation set not used for the final accuracy claim.","section":"Sec. IV.A, Fig. 5"}],"minor_comments":[{"comment":"For the TE polarization, the solved field is H_z, but the captions of Figs. 4 and 7 refer to the plotted field as \"E_zr\" and \"E_zi\" (for example, \"(a) real part E_zr for E_z\"). This is inconsistent with the formulation in Sec. III, where E_z is the nonzero component only in the TM case, and should be corrected.","section":"Sec. IV.B, Figs. 4 and 7"},{"comment":"The text mentions a \"dynamic weighting mechanism\" for lambda_PDE and lambda_BC but never specifies the update rule or the final weights. Without this information, the training procedure is not fully reproducible.","section":"Sec. II, Eq. (3) and Sec. IV"},{"comment":"The caption of Fig. 3 lists horizontal cutlines at \"y=-0.8, 0, and 0.4\" (which appears to list three values), while the main text says the horizontal lines are at y=0.1, 0.5, 0.7, 0.9. The positions and number of cutlines should be made consistent.","section":"Sec. IV.A, Fig. 3 caption and text"},{"comment":"The definition of the signed distance s is ambiguous: the text says \"s>=0 (s<0) outside (inside)\", which can be read as two separate conditions. The authors should state unambiguously that s>0 outside the scatterer and s<0 inside.","section":"Sec. III, Eq. (31)"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the sign error in Eq. (26) is most likely a typographical error in the text rather than an error in the actual implementation, since the reported TM accuracies are consistent with a correct radiation condition. The authors should be asked to correct the equation, provide the quantitative TE errors after smoothing, and specify the dynamic weighting scheme. The journal may also wish to encourage the authors to share code or provide a detailed reproducibility statement, given that no code or data are currently included."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's the short version: the PINN scattering framework in this paper is plausible and the TE smoothing idea is worth knowing, but the printed boundary residual in Eq. (26) enforces the wrong radiation condition at x_min and y_min. With σ = -1, the residual drives ∂u/∂x = +i k0 u at the left edge instead of the stated ∂u/∂x = -i k0 u, i.e., an incoming wave. The stress-test note is right. The fix is simple (σ should be +1 on all four edges when ∂n is outward), but this is the only algorithmic statement of the ABC, so a literal implementation would not produce the reported L2 errors around 0.07–0.15.\n\nCredit where due: the derivation of the TM and TE scattered-field equations is internally consistent, and the identification of the TE singularity from ∇·(1/ε_r ∇) is correct. The tanh smoothing is a sensible remedy, supported by the Fig. 5 parameter study. The four-case validation against analytic/FDTD is a legitimate effort, especially for TM.\n\nSoft spots, in descending order: (1) the BC sign error, which is load-bearing; (2) no code or data, so the numerical claims cannot be independently checked; (3) the smoothing width δ is tuned against the benchmark error—they say \"parameter values that minimize ε_L2\"—so the method has a fitted free parameter without a clear selection rule; (4) the TE arbitrary-structure case lacks a quantitative error; and (5) no error bars anywhere. These are all fixable in revision.\n\nIf the sign error is a typo, the paper is a solid incremental contribution to PINN-based scattering. As printed, it is not reproducible. Send it to peer review, not desk reject: the topic is active, the core ideas are coherent, and the errors are correctable. Reviewers should ask for a corrected BC, a reproducible implementation (code or data), and TE errors for the arbitrary structure. The audience is photonics researchers working on PINN solvers; they will benefit from the TE singularity discussion, but should not trust the reported numbers until the BC is fixed.\n\nRecommendation: accept for peer review with major revision requested.","headline":"The PINN scattering formulation is plausible and the TE smoothing idea is worth knowing, but the printed boundary residual in Eq. (26) enforces the wrong radiation condition, so the reported accuracies are not reproducible as written.","tokens_in":15679,"tokens_out":6828,"would_cite":false,"duration_ms":50375,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A meshless physics-informed neural network, with Maxwell residuals and a first-order radiation condition in its loss, predicts 2D scattered fields for TM and TE polarizations, with TE accuracy restored by hyperbolic-tangent smoothing of…","keywords":["physics-informed neural networks","electromagnetic scattering","Helmholtz equation","TM polarization","TE polarization","radiation boundary condition","hyperbolic tangent smoothing","meshless methods"],"falsifier":"Take an exact outgoing solution, for instance the field of a line source, $u_{\\mathrm{sc}}=H_0^{(1)}(k_0r)$, and evaluate the boundary residual (26) at a point on $x=x_{\\min}$ and at a point on $y=y_{\\min}$. If the residual is close to zero when $u_{\\mathrm{sc}}=0$ but not when the radiating field is used, the boundary loss is not enforcing the stated radiation condition, and the reported errors cannot follow from the printed equations.","tokens_in":14510,"feed_emoji":"📡","tokens_out":15008,"duration_ms":122390,"temperature":0.7,"pith_summary":"This paper aims to show that a physics-informed neural network can act as a general, meshless solver for two-dimensional plane-wave scattering in arbitrary dielectric structures. The authors encode the frequency-domain Maxwell equations and a first-order radiation boundary condition directly into the network's loss, so the network learns the scattered field from the physics alone. For TM polarization, they report relative $L^2$ errors at or below roughly 0.1 for cylinders, concentric shells, and clusters of cylinders, with only slight growth as the geometry becomes more complex. For TE polarization, where the governing equation becomes singular at sharp dielectric interfaces, they introduce a hyperbolic-tangent smoothing of the permittivity jump and report that the predictions then closely match FDTD simulations. The payoff, if the claim holds, is a single training framework that handles both polarizations and irregular geometries without mesh generation or per-geometry solver redesign.","feed_headline":"Neural network solves 2D scattering without a mesh","feed_subtitle":"Handles TM and TE polarizations on cylinders, shells, and irregular dielectric shapes.","key_machinery":"The machine is the composite loss function of a fully connected network with tanh activations, $L=\\lambda_{\\mathrm{PDE}}L_{\\mathrm{PDE}}+\\lambda_{\\mathrm{BC}}L_{\\mathrm{BC}}$. $L_{\\mathrm{PDE}}$ averages the squared residuals of the scattered-field Helmholtz equations, Eq. (17) for TM and Eq. (18) for TE; the TE operator contains the singular term $\\nabla\\cdot(\\varepsilon_r^{-1}\\nabla)$, and the right-hand sides are the equivalent sources (19)--(20). $L_{\\mathrm{BC}}$ averages the squared residuals of the first-order radiation condition, written uniformly for all four edges with the sign variable $\\sigma$ in Eq. (26). The $\\tanh$ permittivity smoothing (31) replaces the discontinuous $\\varepsilon_r$ by a continuous profile of width $\\delta$, eliminating the interface singularities that otherwise stall TE training.","core_discovery":"The central claim is that a single meshless network, trained solely on residuals of the frequency-domain Maxwell equations and a first-order radiation boundary condition, can reproduce the scattered field $u_{\\mathrm{sc}}$ (with $u_{\\mathrm{sc}}=E_z$ for TM and $H_z$ for TE) for dielectric geometries ranging from one cylinder to arbitrary composites. The authors derive from the Helmholtz equation a scattered-field equation whose right-hand side is an equivalent source set by the permittivity contrast, and they minimize the squared residuals of this equation together with the boundary residuals. They report that for TM, the relative $L^2$ error is typically $0.07$--$0.15$, growing only mildly with complexity; for TE, the interface singularity in $\\nabla\\cdot(\\varepsilon_r^{-1}\\nabla)$ degrades accuracy unless the permittivity jump is smoothed with a $\\tanh$ profile, after which the fields closely match FDTD. The paper frames the result as a demonstration that the same training protocol and architecture remain stable and accurate across all tested scatterer configurations.","pith_inferences":["Beyond the paper, the printed boundary residual (26) matters for reproducibility: if the signs at the left and bottom edges are taken literally, the residual vanishes when the scattered field is zero rather than when the radiating condition holds, so reimplementing from the equations may require an unstated sign correction.","The optimal smoothing width likely depends on geometry: the three-cylinder sweep in Fig. 5 favors $\\delta\\gtrsim5\\times10^{-3}$, while the irregular composite is run at $\\delta=10^{-3}$, so a per-geometry $\\delta$ scan would settle whether TE accuracy is robust or tuned.","If the framework is correct, automatic differentiation gives gradients of the predicted field with respect to geometry and permittivity parameters, which could turn this forward solver into a differentiable surrogate for inverse design of dielectric scatterers.","A concrete next test would be to swap the first-order absorbing condition for a perfectly matched layer or higher-order radiation condition and compare $L^2$ errors; that isolates how much of the total error comes from the boundary residual versus the interior Helmholtz residual."],"forward_implications":["With the same architecture and loss recipe, TM relative $L^2$ errors remain at or below about $0.1$ across single, core-shell, three-cylinder, and irregular composite scatterers, so the method transfers across these geometries without re-meshing.","The hyperbolic-tangent smoothing of $\\varepsilon_r$ removes the TE interface singularity, reducing errors from values above $0.9$ to close agreement with FDTD benchmarks.","The dynamic weighting of PDE and boundary losses, together with the dual early-stopping criterion, keeps training stable up to $5\\times10^4$ iterations across all tested configurations.","Because the permittivity enters only as a function $\\varepsilon_r(x,y)$, any new scatterer shape is handled by resampling collocation points and retraining, with no mesh regeneration step."],"supporting_citations":[{"why":"Establishes the physics-informed neural network paradigm of embedding PDE residuals into the loss, which this paper adapts to frequency-domain scattering.","marker":"[13]"},{"why":"Supplies the hyperbolic tangent activation used for all networks, chosen for boundedness and continuous second derivatives.","marker":"[27]"},{"why":"Provides the general PINN loss formulation that motivates the weighted PDE plus boundary residual objective.","marker":"[28]"},{"why":"Supports the use of a finite computational domain with a radiation boundary condition instead of an infinite domain.","marker":"[24]"},{"why":"Also cited for the finite-domain treatment that justifies the local absorbing boundary condition.","marker":"[29]"},{"why":"The FDTD method used as the benchmark for the irregular-scatterer configurations.","marker":"[8]"}],"fun_headline_variants":["No mesh needed: neural net solves 2D scattering","Meshless network predicts waves through arbitrary dielectrics","Physics-informed network handles TM and TE scattering","From one cylinder to composite shapes: meshless solver","Neural net without grid simulates complex EM scattering"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central assumption is that the printed boundary formula (26) actually lets scattered waves leave the computational box; if the signs at the left and bottom edges are taken at face value, those edges force the scattered field to zero, so the reported accuracy relies on an unstated correction.","fun_headline_variants_meta":{"raw":{"variants":["No mesh needed: neural net solves 2D scattering","Meshless network predicts waves through arbitrary dielectrics","Physics-informed network handles TM and TE scattering","From one cylinder to composite shapes: meshless solver","Neural net without grid simulates complex EM scattering"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00035,"raw_usage":{"total_tokens":1917,"prompt_tokens":960,"completion_tokens":957,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":576,"completion_tokens_details":{"reasoning_tokens":882}},"tokens_in":576,"tokens_out":957,"duration_ms":8321,"temperature":1.0,"reasoning_tokens":882,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:24:55.552168+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an exact outgoing solution, for instance the field of a line source, $u_{\\mathrm{sc}}=H_0^{(1)}(k_0r)$, and evaluate the boundary residual (26) at a point on $x=x_{\\min}$ and at a point on $y=y_{\\min}$. If the residual is close to zero when $u_{\\mathrm{sc}}=0$ but not when the radiating field is used, the boundary loss is not enforcing the stated radiation condition, and the reported errors cannot follow from the printed equations.","supporting_citations":[{"cited_title":"Self-scalable Tanh (Stan): Faster Convergence and Better Generalization in Physics-informed Neural Networks","cited_arxiv_id":"2204.12589","evidence_quote":"Supplies the hyperbolic tangent activation used for all networks, chosen for boundedness and continuous second derivatives."},{"cited_title":"Scientific machine learning through physics-informed neural networks: Where we are and what’s next,","cited_arxiv_id":null,"evidence_quote":"Provides the general PINN loss formulation that motivates the weighted PDE plus boundary residual objective."},{"cited_title":"Physics-informed deep learning for 3D modeling of light diffraction from optical metasurfaces,","cited_arxiv_id":null,"evidence_quote":"Supports the use of a finite computational domain with a radiation boundary condition instead of an infinite domain."},{"cited_title":"A novel 3-D hybrid approach for simulating electromagnetic scattering from electrically large targets in ducting maritime environments,","cited_arxiv_id":null,"evidence_quote":"Also cited for the finite-domain treatment that justifies the local absorbing boundary condition."},{"cited_title":"The sensitivity of InSAR closure phase to spatial variations of soil structure and moisture as revealed by FDTD simulations,","cited_arxiv_id":null,"evidence_quote":"The FDTD method used as the benchmark for the irregular-scatterer configurations."}],"review_version":1}