Organizations: Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven, The Netherlands · Signify, High Tech Campus 7, 5656 AE, Eindhoven, The Netherlands
We address the inverse problem of designing two-dimensional reflectors that transform light from a finite, extended source into a prescribed far-field distribution. The reflector height is represented by a neural network and optimized with two objective functions: a direct change-of-variables loss based on the closed-form inverse ray map, and a mesh-based loss that maps target cells back to the source and remains usable for discontinuous sources. Gradients are computed by automatic differentiation and minimized with a robust quasi-Newton method. As a baseline, we adapt a deconvolution pipeline built on a simplified finite-source approximation: a one-dimensional monotone map is recovered from flux balance, converted to a reflector by an integrating-factor ODE solve, and embedded in a modified Van Cittert iteration with nonnegativity clipping and ray-traced feedback. Across four benchmarks, covering continuous and discontinuous sources and minimum-height constraints, accuracy is measured by ray-traced normalized mean absolute error. On the two main benchmarks, the neural method reaches errors of about 2e-5 and 5e-5 within a few seconds on one NVIDIA RTX 4090 GPU, compared with 4e-3 and 5e-2 for the deconvolution baseline after several hundred seconds. The results show that the neural formulation is both more accurate and substantially faster, while still supporting practical height constraints. We also discuss extensions to rotationally symmetric and full three-dimensional reflector design through iterative correction schemes.
Figures & tables
Figure 1: Finite-source-to-far-field reflector system setup.
Figure 2: Finite-source-to-far-field reflector system approximation setup.
Symbol
Meaning
Ω
Spatial support of the linear light source (source base domain)
A
Angular source domain (radian angles from positive x -axis)
S
Full source domain, S:=Ω×A
Σ
Stereographic far-field target domain
T
Full target domain, T:=Ω×Σ
f(s,α)
Source distribution function over S
Table 1: Symbols used throughout the paper.
Figure 3: Mesh-based procedure for computing a continuous loss for inverse reflector design. (a) The target space is discretized into quadrilateral cells. (b) Each vertex is mapped to the source space. (c) The source distribution is evaluated. (d) Integration is performed over valid intersections with the source domain. (e) The integrals are assigned back to the original mesh cells. (f) A histogram of the far-field output is obtained.
Figure 4: Reflectors for the mapping m(s)=(2s+s3)/3 for different values of h computed using Eq. ( 35 ). The left plot shows the mapping, the center plot shows the obtained functions u for different values of h , and the right plot shows the corresponding reflectors in the xz -plane.
Figure 5: Ground-truth reflector, source distribution, corresponding target distribution, and far-field target distribution for Example A .
Figure 6: Final reflectors, pointwise absolute reflector error, and convergence history for the neural-network and deconvolution methods applied to Example A . In (a) the ground-truth reflector is drawn as a dashed black curve on top of the predictions so that the otherwise-coincident neural-network curve remains visible. Panel (b) gives the pointwise absolute deviation from ground truth.
Figure 7: Ground-truth reflector, source distribution, corresponding target distribution, and far-field target distribution for Example B .
Figure 8: Final reflectors, pointwise absolute reflector error, and convergence history for the neural-network and deconvolution methods applied to Example B . In (a) the ground-truth reflector is drawn as a dashed black curve on top of the predictions so that the otherwise-coincident neural-network curve remains visible. Panel (b) gives the pointwise absolute deviation from ground truth.
Example
Method
Best RT NMAE
Time-to-best
Iterations
Final loss
A
NN
2.16×10−5
0.34 s
209
1.29×10−11
A
Deconv
4.44×10−3
410 s
900
—
B
NN
4.65×10−5
2.54 s
278
6.74×10−12
B
Deconv
5.02×10−2
611 s
3000
—
Table 2: Summary of accuracy and computational cost for the neural-network (NN) and deconvolution (Deconv) methods on Example A and Example B . “Best RT NMAE” is the smallest ray-traced NMAE attained over the run, “time-to-best” is the wall-clock time at which that minimum is reached, “Iterations” is the corresponding number of accepted/applied updates, and “Final loss” is the value of L(θ) at the end of training for the neural method (not defined for the deconvolution baseline). All runs are on a single NVIDIA RTX 4090 GPU.
Figure 9: NMAE for the neural-network and deconvolution methods as a function of minimum reflector height hmin for Example C .
Figure 10: NMAE for the neural-network and deconvolution methods as a function of minimum reflector height hmin for Example D .
Figure 11: Hyperparameter sensitivity for the direct change-of-variables method on Example A : ray-tracing NMAE as a function of optimization time for one-factor sweeps over (a) MLP depth, (b) MLP width, (c) far-field sample count nσ , and (d) p -sample count np (five settings × five seeds each). Solid curves are per-setting means in log space; shaded bands are 95 % confidence intervals. The dashed line is the single-run baseline from Figure 6 .
Figure 12: Hyperparameter sensitivity for the mesh-based method on Example B : ray-tracing NMAE as a function of optimization time for one-factor sweeps over (a) MLP depth, (b) MLP width, (c) σ -mesh resolution nσmesh , and (d) p -mesh resolution npmesh (five settings × five seeds each). Solid curves are per-setting means in log space; shaded bands are 95 % confidence intervals. The dashed line is the single-run baseline from Figure 8 .
We present a direct optimization method for three-dimensional freeform reflectors that transform the light of a finite-étendue source into a prescribed far-field angular intensity distribution. The reflector profile is represented by a small neural network (a multilayer perceptron), which is trained end-to-end through a differentiable ray-tracing objective. We furthermore parameterize the emission directions in gnomonic coordinates, and show how we use this to ensure that every emitted ray intersects the reflector. At each iteration, the network is converted to a bicubic spline representation for ray-tracing efficiency, and intersections with this smooth surface are solved by a damped Newton solve, with gradients computed via the implicit function theorem. The traced output distribution is compared with the desired target on a 'soft' histogram, under an H−1-type spectral weighting that emphasizes long-range transport of flux to improve convergence. Optimization is performed using a BFGS method with self-scaled Broyden updates and a plateau-perturbation rule to prevent stalling. The method converges reliably within seconds on a single GPU for all examples tested.
Roel Hacking, Lisa Kusch, Martijn Anthonissen +1
Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands · Signify, High Tech Campus 7, 5656 AE Eindhoven, The Netherlands
Neural network surrogates are an emerging alternative to traditional electromagnetic wave simulators like finite-difference time-domain (FDTD); their goal is to replace rigorous physical simulations with pre-trained neural networks that solve wave-scattering forward and inverse problems orders of magnitude faster. However, nonrecurrent, single-step surrogates have scaled only to a few tens of simulation variables. Here, we show that this barrier can be overcome by dynamically generating salient training examples during training, rather than randomly sampling the large space of possible examples. We introduce an algorithm that runs in parallel with surrogate training, using gradient ascent to search refractive-index and source configurations for cases where the surrogate disagrees with a full-wave ground-truth simulator. We also use source and ground-truth normalization with an evolving replay dataset to stabilize and accelerate learning. Using this approach, we train a fast, single-step surrogate for two-dimensional wave scattering with up to 41,772 controllable variables, including dense, freely configurable grids of refractive indices and complex-valued sources. The resulting neural surrogate is robustly accurate across diverse structured and unstructured examples and generalizes inductively to larger domains, reaching over 3 million controllable variables without retraining, a 73.8× increase. We demonstrate the surrogate on large-scale forward simulations and inverse design of freeform beam splitters and gradient-index (GRIN) lenses up to 98 wavelengths wide, showing comparable or better performance than FDTD-based designs, with speedups from 1.29× to 26.5×. These results demonstrate a practical path toward fast, robustly accurate, inductively scalable neural simulators for photonic inverse design and other wave-scattering inverse problems.
Charles Dove, Laura Waller
Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, Berkeley, California 94720, United States
Meta-optics promises compact, high-performance imaging and color routing. However, designing high-performance structures is a high-dimensional optimization problem: mapping a desired optical output back to a physical 3D structure requires solving computationally expensive Maxwell's equations iteratively. Even with adjoint optimization, broadband design can require thousands of Maxwell solves, making industrial-scale optimization slow and costly. To overcome this challenge, we propose the Neural Adjoint Method, a solver-supervised surrogate that predicts 3D adjoint gradient fields from a voxelized permittivity volume using a Fourier Neural Operator (FNO). By learning the dense, per-voxel sensitivity field that drives gradient-based updates, our method can replace per-iteration adjoint solves with fast predictions, greatly reducing the computational cost of full-wave simulations required during iterative refinement. To better preserve sensitivity peaks, we introduce a stage-wise FNO that progressively refines residual errors with increasing emphasis on higher-frequency components. We curate a meta-optics dataset from paired forward/adjoint FDTD simulations and evaluate it across three tasks: spectral sorting (color routers), achromatic focusing (metalenses), and waveguide mode conversion. Our method reduces design time from hours to seconds. These results suggest a practical route toward fast, large-scale volumetric meta-optical design enabled by AI-accelerated scientific computing.
Chanik Kang, Hyewon Suk, Haejun Chung
Hanyang University, Department of Artificial Intelligence Semiconductor Engineering Seoul, Korea · Hanyang University, Department of Electronic Engineering Seoul, Korea