Physics-Informed Neural Networks for Optimal Beam Shaping in Flat Optics

Rafael de la Fuente Herrezuelo

REVIEW 3 major objections 4 minor 10 references

A physics-informed neural network solves the Monge-Ampère beam-shaping equation for flat optics, producing phase masks that outperform conventional phase retrieval on a structured far-field target.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-01 16:21 UTC pith:QXMP5WRO

load-bearing objection Legitimate new PINN application with real diffraction validation, but the Monge–Ampère solver omits the boundary condition that makes the transport map onto, and the GS comparison is not convergence-checked. the 3 major comments →

arxiv 2607.18012 v2 pith:QXMP5WRO submitted 2026-07-20 physics.optics physics.comp-ph

Physics-Informed Neural Networks for Optimal Beam Shaping in Flat Optics

Rafael de la Fuente Herrezuelo This is my paper

classification physics.optics physics.comp-ph

keywords physics-informed neural networksbeam shapingflat opticsphase masksMonge-Ampère equationray mappingscalar diffractionmetasurfaces

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a physics-informed neural network (PINN) that designs phase-only optical elements for beam shaping by solving the Monge-Ampère ray-mapping equation. The trained network outputs a smooth phase profile that, when validated by scalar diffraction, reproduces a prescribed target intensity with a relative RMS error of about 2.97% for a far-field logo pattern, compared with 22.6% for a standard phase-retrieval algorithm. This matters because beam shaping is used in laser machining, displays, and flat optics, and the PINN avoids the speckle and irregular phases typical of Fourier-based methods. The approach extends previous cylindrically symmetric metasurface designs to arbitrary two-dimensional targets without relying on data-driven training.

Core claim

The central claim is that a PINN can directly solve the Monge-Ampère equation that governs energy-conserving ray mappings for a phase-only optical element, for both finite-distance and far-field targets. The network represents the phase potential and is trained by minimizing the residual of the PDE, using automatic differentiation to compute the required second derivatives. For a Gaussian beam to a 'DT' logo target in the far field, the learned phase produces an energy efficiency of about 99.98% inside the target support, versus 95.90% for a conventional projection-based algorithm, and suppresses the high-frequency speckle caused by irregular phase retrieval. The ray-based design is confirme

What carries the argument

The key machinery is the Monge-Ampère equation derived from the generalized Snell law and power conservation. For far-field targets, the equation reduces to a standard Hessian-determinant form in which the phase gradient directly gives the direction cosines of the target; for finite distances, additional terms involving the ray propagation length and the longitudinal component K appear. The field is represented by a fully connected neural network with smooth tanh activations, and automatic differentiation provides all derivatives needed for the residual. The target intensity is sampled at the phase-dependent mapped coordinates using differentiable bilinear resampling, so the gradients flow t

Load-bearing premise

The method assumes that satisfying the interior Monge-Ampère residual alone is enough to produce a globally energy-conserving ray map, without an explicit boundary condition linking source aperture to target boundary; if this fails, parts of the target may be missing or under-illuminated.

What would settle it

A direct test would be to design a target distribution concentrated near the edge of the aperture or with a sharp cutoff, and measure via wave propagation whether the PINN's phase produces the correct boundary coverage and total energy. A failure would show missing or extra energy at the edges, or an efficiency below the reported 99.98%.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

If the method holds, designing phase masks for complicated two-dimensional beam shapes becomes a PDE-solving task rather than an iterative Fourier projection, potentially yielding smoother and more accurate optic surfaces.
The approach can be extended to non-cylindrically symmetric targets, opening up uses in structured illumination, holography, and display backlighting.
The ray-based PINN solution remains valid under scalar wave propagation for the tested cases, suggesting that the method bridges geometric and wave descriptions of beam shaping.
Because the training signal is the physics residual itself, the method requires no labeled phase data and can be adapted to new source-target pairs by retraining the network for that specific design.
The reported efficiency and error metrics imply that phase-only elements can channel nearly all incident power into a desired pattern, which is valuable for high-power laser applications.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The absence of an explicit boundary condition enforcing that the source aperture maps onto the entire target domain is a likely weakness. Without such a condition, the interior residual may leave portions of the target unilluminated or violate global energy conservation, even if post-hoc admissibility checks catch gross errors.
A natural testable extension is to add a boundary loss or use a conservative formulation; if successful, this would make the method robust for targets whose support is strongly clipped by the aperture.
The same residual approach could be combined with a parameterized target family to build a generative model for beam-shaping optics, though such a model would require many training runs and may struggle with very sharp or discontinuous target features.
The method's reliance on smooth tanh activations and a scalar phase potential may limit its ability to represent wavefronts with vortices or other phase singularities, which are outside the Monge-Ampère transport-map description.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Legitimate new PINN application with real diffraction validation, but the Monge–Ampère solver omits the boundary condition that makes the transport map onto, and the GS comparison is not convergence-checked.

read the letter

First thing to know: this is a genuinely new application — a PINN solving the Monge–Ampère ray-mapping equation for flat-optics beam shaping, including non-axisymmetric targets, with independent scalar diffraction validation for both examples. The code and data are on GitHub. That puts it ahead of many papers in this area.

What it does well: the equations are derived carefully from the generalized Snell law, the normalization is sensible, and the network is trained entirely on the PDE residual, so no circularity from data labels. The far-field DT logo is compared against Gerchberg–Saxton, and the PINN phase indeed produces lower RMS error (2.97% vs 22.6%) and visibly less speckle. The independent diffraction check is the right validation and it passes.

The soft spot is exactly what the stress-test note flags. The Monge–Ampère formulation is an optimal transport problem, and without a boundary condition mapping the source aperture boundary onto the target boundary, the solution can be non-onto — some target regions unilluminated, some source power double-counted. The paper explicitly imposes no Dirichlet condition, only post-hoc checks: mapped points inside the target, K real. Those checks are local; they do not enforce global flux balance or target coverage. So the PDE formulation is structurally incomplete. The diffraction simulations show the method worked for the two tested targets, which suggests the network implicitly found a boundary-satisfying solution. But "happened to" is not a method, and the general claim that this solves the beam-shaping problem needs qualification.

Second point, minor: the GS comparison runs only 60 iterations with no convergence check. That makes the quantitative superiority claim suggestive, not conclusive. The reported energy efficiency of 99.98% also looks too clean, though it is not central.

Citations: the paper cites the author's own project report [2] for the standard ray-mapping equation. That is not ideal — a proper optics reference would be better — but it is not what drives the result, and the independent diffraction validation breaks any circularity concern.

Bottom line: honest work, real examples, a correctable but real technical gap. The missing boundary condition should be fixed before the method is claimed to solve general beam shaping, but for the demonstrated targets the claim holds. I would send this to peer review and ask for the boundary-condition fix and a stronger GS baseline.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a physics-informed neural network (PINN) approach for designing phase-only flat-optics elements that reshape an input beam into prescribed intensity distributions. The phase profile is represented by a fully connected network and trained by minimizing residuals of Monge–Ampère equations derived from generalized-Snell-law ray mappings, for both finite-distance targets (Eq. 29) and far-field angular targets (Eq. 37). The learned phases are validated with independent scalar diffraction simulations for a Gaussian-to-star finite-distance example and a Gaussian-to-logo far-field example, and the far-field result is compared against 60 iterations of the Gerchberg–Saxton algorithm. The central claims are that this is the first PINN application to flat-optics beam shaping and that it achieves lower error and higher energy efficiency than GS for the tested target.

Significance. If fully supported, the method would be a useful addition to flat-optics design: it replaces iterative phase retrieval with a PDE-constrained neural optimization, handles non-cylindrically symmetric targets, and the authors provide open code and independent diffraction validation. The ray-mapping derivation appears self-consistent, and the normalization of the equations is nontrivial and largely convincing. However, the central claim is currently weakened by a structural gap in the PDE formulation: no boundary condition is imposed on the Monge–Ampère equation. Because the method is demonstrated on only one target per geometry and the GS baseline is not shown to be converged, the generality and superiority claims are not yet established at the level required for publication.

major comments (3)

[§3.2, Eqs. (29), (37)] The Monge–Ampère equation is solved without any boundary condition. The manuscript explicitly states that no Dirichlet condition is imposed and that admissibility is only checked after training by verifying that mapped points remain in the target domain and that K is real. This is not sufficient: in optimal transport, the correct boundary condition—typically that the source aperture boundary maps onto the target boundary—is required for the solution to be a genuine transport map that covers the full target and conserves energy globally. A network can minimize the interior residual in Eq. (41) while producing a ray map that is not onto, leaving target regions unilluminated or delivering source power multiple times to the same region. The post-hoc checks are local admissibility checks, not global flux-balance checks. The reported 99.98% energy efficiency comes from wave-optics simulations
[§4.3] The comparison with Gerchberg–Saxton is weakened by the use of only 60 iterations with no reported convergence criterion. GS is known to converge slowly for structured targets such as a two-letter logo, so an underconverged baseline is not a fair reference for the claimed superiority (relative RMS 2.97% vs 22.6%, energy efficiency 99.98% vs 95.90%). Please report the error as a function of iteration count and either run GS until a convergence metric stagnates or state the stopping rule. Without this, the quantitative superiority claim is not decisive.
[§4.1–§4.2, §5] The paper's general claim—that the PINN approach solves beam-shaping problems in flat optics—rests on only two numerical examples, one per formulation. While the finite-distance star example is smooth and the far-field logo example has internal strokes, a single realization per geometry is not enough to establish that the method reliably produces admissible, boundary-satisfying transport maps, especially given the missing boundary condition. I recommend adding at least one additional target per formulation, or tempering the conclusions to a proof-of-concept demonstration. This is load-bearing for the stated scope of the method.

minor comments (4)

[§3.2] The notation D_near in Eq. (39) is introduced without a formal definition. It is presumably the left-hand side of Eq. (29) before multiplying by K^4; please define it explicitly to avoid ambiguity.
[§4.3] The MSE values (3.59e6 and 2.08e8) are reported without normalization or units. Please report per-pixel or normalized MSE and specify the integration domain, so the comparison is reproducible.
[Fig. 5] The panels in Fig. 5 are not individually labeled or captioned. Adding labels (a)–(d) or similar would greatly improve readability, especially since the figure is central to the validation claim.
[References] The derivation of the Monge–Ampère beam-shaping equation cites the author's own project report [2] and a PhD thesis [3]. For a journal submission, please also cite peer-reviewed sources for the ray-mapping/Monge–Ampère formulation, e.g., standard references on nonimaging optics or phase retrieval.

Circularity Check

0 steps flagged

No significant circularity: the PINN phase design is trained on a physics-based Monge–Ampère residual and validated by independent scalar diffraction, so the central claim is not equivalent to its inputs.

full rationale

The paper’s derivation chain is self-contained. It starts from the thin-element phase ansatz in Eq. (1), the generalized Snell law in Eqs. (3)–(4) (an external result from Yu et al., [8]), and the geometric ray map in Eqs. (8)–(9). Power conservation in Eqs. (10)–(11) yields the Jacobian condition, which is expanded into the finite-distance Monge–Ampère equation Eq. (13) and the far-field Monge–Ampère equation Eq. (24). The target intensity E or I_e,Ωcosθ is a prescribed input, not an output of the network; the free function is the phase Φ, initialized with no phase labels or precomputed ray correspondences. The loss Eq. (41) penalizes the residual of the PDE only, with automatic differentiation providing derivatives. The resulting phase is then checked by scalar wave propagation using diffractsim, an independent forward model. No step reduces the claimed prediction—the phase mask—to the target input by construction. Several references are to the author’s own prior work ([2], [6], [7], [9]), but these are contextual or supporting citations: the Monge–Ampère equation is re-derived explicitly rather than imported as an unverified black box, and the prior metasurface demonstrations are cited only as motivation for extending to non-cylindrical targets. The absence of an explicit transport boundary condition is a genuine mathematical well-posedness concern and could affect the generality of the method, but it does not make the derivation circular: it is a structural gap in the PDE formulation, not an identity between inputs and outputs. The paper therefore exhibits no significant circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 4 axioms · 0 invented entities

The central claim rests on standard physical assumptions (thin element, scalar diffraction, single-valued ray map) and a standard machine-learning assumption (PINN convergence). The only ad hoc free parameter is the phase scale sigma_Phi, which affects training but not the physics. No new physical entities are introduced.

free parameters (1)

sigma_Phi = not specified (chosen via Bayesian optimization)
Controls the phase-gradient scale in the PINN. Too large violates the propagating-ray condition (K real). Selected based on validation loss and map admissibility (Section 3.2). This is a hand-tuned training hyperparameter.

axioms (4)

domain assumption Thin-phase element: the output field is input amplitude times phase factor (amplitude unchanged immediately after surface).
Used in Eq. (1) to model the optical element. Standard in flat optics but not valid for all metasurfaces (e.g., where amplitude also varies).
domain assumption Energy-conserving, single-valued ray map: no caustics or folds.
The Monge-Ampère equation (Eqs. 13, 24) requires the ray mapping to be one-to-one. Folds would invalidate the local conservation law. The paper notes this in Section 2.1 but assumes it holds for the tested targets.
domain assumption Scalar diffraction theory (angular-spectrum) is valid for validation.
Independent validation uses scalar diffraction simulations via diffractsim. This assumes a scalar field and appropriate propagation model, which is typical for the aperture sizes and wavelengths used.
standard math The PINN can approximate the PDE solution and training converges.
Relies on universal approximation and successful training of PINNs. Not formally proven in the paper.

pith-pipeline@v1.3.0-alltime-deepseek · 5929 in / 12100 out tokens · 119670 ms · 2026-08-01T16:21:08.798296+00:00 · methodology

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Cite this review

Pith. "Pith review of Physics-Informed Neural Networks for Optimal Beam Shaping in Flat Optics." pith.science (2026). https://pith.science/paper/QXMP5WRO

@misc{pith2026260718012,
  author       = {Pith},
  title        = {Pith review of: Physics-Informed Neural Networks for Optimal Beam Shaping in Flat Optics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QXMP5WRO}},
  note         = {Machine review of arXiv:2607.18012}
}

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read the original abstract

We introduce a physics-informed neural network (PINN) approach for designing phase profiles in flat optics that reshape an incident beam into a prescribed target intensity distribution. The method solves Monge--Amp\`ere beam-shaping equations associated with energy-conserving ray mappings generated by a phase-only optical element. We treat both finite-distance and far-field targets using a generalized-Snell-law formulation. The learned phase profiles are validated by scalar diffraction simulations and compared with conventional phase-retrieval methods such as Gerchberg--Saxton. To our knowledge, this is the first time a PINN has been used for beam shaping problems in flat optics.

Figures

Figures reproduced from arXiv: 2607.18012 by Rafael de la Fuente Herrezuelo.

**Figure 1.** Figure 1: Phase modulation for structured-beam generation. The incident field is phase-modulated by an SLM or metasurface and forms the target intensity after propagation. mask the scalar field is approximated by Uout(x, y) ≈ Uin(x, y) exp[iΦ(x, y)]. (1) The input irradiance is I(x, y) = 1 2 c ninϵ0 Uin(x, y) 2 , (2) and the input and target intensities are normalized to equal total power, as required by a phase-on… view at source ↗

**Figure 3.** Figure 3: Far-field geometry. The target is specified as an angular distribution Ie,Ω cos θ(α, β). with A1 = z 2 , (14) A2 = zK 1 − Φ˜ 2 y , (15) A3 = zK 1 − Φ˜ 2 x , (16) A4 = 2zK Φ˜ xΦ˜ y, (17) A5 = K4 . (18) The Hessian determinant in Eq. (13) is the characteristic Monge–Ampere nonlinearity, while the remaining terms arise ` from the finite-distance coordinates and the normalized slopes Φ˜ x/K and Φ˜ y/K.… view at source ↗

**Figure 4.** Figure 4: PINN architecture. The network outputs Φ¯ and automatic differentiation supplies the derivatives entering the Monge–Ampere ` residual. the far-field Monge–Ampere equation is ` Ie,Ω cos θ Φ˜ x, Φ˜ y [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

**Figure 5.** Figure 5: PINN beam-shaping validation. The upper row shows a finite-distance Gaussian-to-star flat-top design. The lower row shows a far-field Gaussian-to-DT-logo design. The panels compare phase profiles, prescribed targets, propagated or far-field simulated irradiances, and central cross sections. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

**Figure 6.** Figure 6: Gerchberg–Saxton comparison for the DT far-field target. Panels (a,b) show the GS phase mask on the full aperture and in a central zoomed region (red box). Panels (c,d) show the corresponding far-field prediction on the full and zoomed angular domains. Panel (e) compares the β = 0 cross sections of the GS prediction, the PINN prediction, and the target profile. 4.3 Comparison with existing methods The DT f… view at source ↗

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 3 canonical work pages

[1]

A practical algorithm for the determination of phase from image and diffraction plane pic- tures,

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pic- tures,”Optik35, 237–246 (1972)

1972
[2]

Design and simulation of structured beams using phase profiles,

Rafael de la Fuente Herrezuelo, “Design and simulation of structured beams using phase profiles,” project report, Univer- sidad Nacional de Educaci´on a Distancia (2025)

2025
[3]

Nielsen, K. E. S. (2024). Optical metasurfaces for lighting ap- plications. Department of Physics, Technical University of Den- mark

2024
[4]

Physics- informed neural networks: A deep learning framework for solv- ing forward and inverse problems involving nonlinear partial differential equations,

M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics- informed neural networks: A deep learning framework for solv- ing forward and inverse problems involving nonlinear partial differential equations,”Journal of Computational Physics378, 686–707 (2019). doi:10.1016/j.jcp.2018.10.045

doi:10.1016/j.jcp.2018.10.045 2019
[5]

Physics-informed neural networks for inverse problems in nano-optics and metamaterials,

Y . Chen, L. Lu, G. E. Karniadakis, and L. Dal Negro, “Physics-informed neural networks for inverse problems in nano-optics and metamaterials,”Optics Express28(8), 11618– 11633 (2020). doi:10.1364/OE.384875

doi:10.1364/oe.384875 2020
[6]

Cylindrically-symmetric collimated beam shaping using non- imaging metasurfaces,

Kirstine E. S. Nielsen, Xavier Zambrana-Puyalto, Rafael de la Fuente Herrezuelo, Villads Egede Johansen, and Søren Raza, “Cylindrically-symmetric collimated beam shaping using non- imaging metasurfaces,”Proc. SPIE13132, 1313208 (2024). doi:10.1117/12.3027738

work page doi:10.1117/12.3027738 2024
[7]

Experimental demonstration of a beam shaping non- imaging metasurface,

Kirstine E. S. Poulsen, Xavier Zambrana-Puyalto, Rafael de la Fuente Herrezuelo, Villads Egede Johansen, and Søren Raza, “Experimental demonstration of a beam shaping non- imaging metasurface,”Optics Express33(9), 19119–19129 (2025). doi:10.1364/OE.559542

work page doi:10.1364/oe.559542 2025
[8]

Light propagation with phase discon- tinuities: generalized laws of reflection and refraction,

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Ca- passo, and Z. Gaburro, “Light propagation with phase discon- tinuities: generalized laws of reflection and refraction,”Science 334(6054), 333–337 (2011). doi:10.1126/science.1210713

doi:10.1126/science.1210713 2011
[9]

diffractsim: A flex- ible Python diffraction simulator,

R. de la Fuente Herrezuelo, “diffractsim: A flex- ible Python diffraction simulator,” GitHub (2022). doi:10.5281/zenodo.6147771

work page doi:10.5281/zenodo.6147771 2022
[10]

Spatial Transformer Networks,

M. Jaderberget al., “Spatial Transformer Networks,”Ad- vances in Neural Information Processing Systems28, 2017– 2025 (2015). doi:10.48550/arXiv.1506.02025 5

Pith/arXiv arXiv doi:10.48550/arxiv.1506.02025 2017

[1] [1]

A practical algorithm for the determination of phase from image and diffraction plane pic- tures,

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pic- tures,”Optik35, 237–246 (1972)

1972

[2] [2]

Design and simulation of structured beams using phase profiles,

Rafael de la Fuente Herrezuelo, “Design and simulation of structured beams using phase profiles,” project report, Univer- sidad Nacional de Educaci´on a Distancia (2025)

2025

[3] [3]

Nielsen, K. E. S. (2024). Optical metasurfaces for lighting ap- plications. Department of Physics, Technical University of Den- mark

2024

[4] [4]

Physics- informed neural networks: A deep learning framework for solv- ing forward and inverse problems involving nonlinear partial differential equations,

M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics- informed neural networks: A deep learning framework for solv- ing forward and inverse problems involving nonlinear partial differential equations,”Journal of Computational Physics378, 686–707 (2019). doi:10.1016/j.jcp.2018.10.045

doi:10.1016/j.jcp.2018.10.045 2019

[5] [5]

Physics-informed neural networks for inverse problems in nano-optics and metamaterials,

Y . Chen, L. Lu, G. E. Karniadakis, and L. Dal Negro, “Physics-informed neural networks for inverse problems in nano-optics and metamaterials,”Optics Express28(8), 11618– 11633 (2020). doi:10.1364/OE.384875

doi:10.1364/oe.384875 2020

[6] [6]

Cylindrically-symmetric collimated beam shaping using non- imaging metasurfaces,

Kirstine E. S. Nielsen, Xavier Zambrana-Puyalto, Rafael de la Fuente Herrezuelo, Villads Egede Johansen, and Søren Raza, “Cylindrically-symmetric collimated beam shaping using non- imaging metasurfaces,”Proc. SPIE13132, 1313208 (2024). doi:10.1117/12.3027738

work page doi:10.1117/12.3027738 2024

[7] [7]

Experimental demonstration of a beam shaping non- imaging metasurface,

Kirstine E. S. Poulsen, Xavier Zambrana-Puyalto, Rafael de la Fuente Herrezuelo, Villads Egede Johansen, and Søren Raza, “Experimental demonstration of a beam shaping non- imaging metasurface,”Optics Express33(9), 19119–19129 (2025). doi:10.1364/OE.559542

work page doi:10.1364/oe.559542 2025

[8] [8]

Light propagation with phase discon- tinuities: generalized laws of reflection and refraction,

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Ca- passo, and Z. Gaburro, “Light propagation with phase discon- tinuities: generalized laws of reflection and refraction,”Science 334(6054), 333–337 (2011). doi:10.1126/science.1210713

doi:10.1126/science.1210713 2011

[9] [9]

diffractsim: A flex- ible Python diffraction simulator,

R. de la Fuente Herrezuelo, “diffractsim: A flex- ible Python diffraction simulator,” GitHub (2022). doi:10.5281/zenodo.6147771

work page doi:10.5281/zenodo.6147771 2022

[10] [10]

Spatial Transformer Networks,

M. Jaderberget al., “Spatial Transformer Networks,”Ad- vances in Neural Information Processing Systems28, 2017– 2025 (2015). doi:10.48550/arXiv.1506.02025 5

Pith/arXiv arXiv doi:10.48550/arxiv.1506.02025 2017

Physics-Informed Neural Networks for Optimal Beam Shaping in Flat Optics

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it