ENA: Inverse Design for Optical Multilayers

• The paper introduces an extended neural adjoint (ENA) framework that integrates a forward surrogate, material-aware loss, and regression activation mapping to enhance the inverse design of optical multilayer thin films.
• ENA utilizes a lightweight OMT-FNN architecture that accurately predicts spectral responses, enabling efficient gradient-based optimization over variable-length design spaces.
• F-RAM visualizes layer-wise feature importance to ensure interpretability, demonstrating robust performance and diversity compared to benchmark methods like Res-GLOnet.

Searching arXiv for the main paper and directly related methods mentioned in the provided data. arXiv_search(query="(Kim et al., 10 Jul 2025)", max_results=5, sort_by="submittedDate") to=arXiv_search 天天中彩票篮球 нәтиجے code 200 [{"arxiv_id":"(Kim et al., 10 Jul 2025)","title":"Interpretable inverse design of optical multilayer thin films based on extended neural adjoint and regression activation mapping","authors":["Nayoung Ahn","Jungho Lee","Gwanho Kim","Yu-Jin Heo","Hera Kim"],"abstract":"We propose an extended neural adjoint (ENA) framework, which meets six key criteria for artificial intelligence-assisted inverse design of optical multilayer thin films (OMTs): accuracy, efficiency, diversity, scalability, flexibility, and interpretability. To enhance the scalability of the existing neural adjoint method, we present a novel forward neural network architecture for OMTs and introduce a material loss function into the existing neural adjoint loss function, facilitating the exploration of material configurations of OMTs. Furthermore, we present the detailed formulation of the regression activation mapping for the presented forward neural network architecture (F-RAM), a feature visualization method aimed at improving interpretability. We validated the efficacy of the material loss by conducting an ablation study, where each component of the loss function is systematically removed and evaluated. The results indicated that the inclusion of the material loss significantly improves accuracy and diversity. To substantiate the performance of the ENA-based inverse design, we compared it against the residual network-based global optimization network (Res-GLOnet). The ENA yielded the OMT solutions of an inverse design with higher accuracy and better diversity compared to the Res-GLOnet. To demonstrate the interpretability, we applied F-RAM to diverse OMT structures with similar optical properties, obtained by the proposed ENA method. We showed that distributions of feature importance for various OMT structures exhibiting analogous optical properties are consistent, despite variations in material configurations, layer number, and thicknesses. Furthermore, we demonstrate the flexibility of the ENA method by restricting the initial layer of OMTs to SiO2 and 100 nm."}] Searching for the residual network-based global optimization network mentioned in the data. arXiv_search(query="Res-GLOnet optical multilayer thin films", max_results=10, sort_by="relevance") to=arXiv_search դարձել code 200 [{"arxiv_id":"(Kim et al., 10 Jul 2025)","title":"Interpretable inverse design of optical multilayer thin films based on extended neural adjoint and regression activation mapping","authors":["Nayoung Ahn","Jungho Lee","Gwanho Kim","Yu-Jin Heo","Hera Kim"],"abstract":"We propose an extended neural adjoint (ENA) framework, which meets six key criteria for artificial intelligence-assisted inverse design of optical multilayer thin films (OMTs): accuracy, efficiency, diversity, scalability, flexibility, and interpretability. To enhance the scalability of the existing neural adjoint method, we present a novel forward neural network architecture for OMTs and introduce a material loss function into the existing neural adjoint loss function, facilitating the exploration of material configurations of OMTs. Furthermore, we present the detailed formulation of the regression activation mapping for the presented forward neural network architecture (F-RAM), a feature visualization method aimed at improving interpretability. We validated the efficacy of the material loss by conducting an ablation study, where each component of the loss function is systematically removed and evaluated. The results indicated that the inclusion of the material loss significantly improves accuracy and diversity. To substantiate the performance of the ENA-based inverse design, we compared it against the residual network-based global optimization network (Res-GLOnet). The ENA yielded the OMT solutions of an inverse design with higher accuracy and better diversity compared to the Res-GLOnet. To demonstrate the interpretability, we applied F-RAM to diverse OMT structures with similar optical properties, obtained by the proposed ENA method. We showed that distributions of feature importance for various OMT structures exhibiting analogous optical properties are consistent, despite variations in material configurations, layer number, and thicknesses. Furthermore, we demonstrate the flexibility of the ENA method by restricting the initial layer of OMTs to SiO2 and 100 nm."},{"arxiv_id":"(Polášková et al., 2021)","title":"Global optimization neural network for computational design of photonic structures","authors":["Sang-Hyun Park","Jong Woo You","Bumki Min"],"abstract":"For the inverse design of photonic structures, global optimization methods have proven to be effective but are often computationally expensive. We propose a global optimization neural network (GLOnet), a generative neural network designed to optimize device topologies globally over the entire design space with one training process. Our GLOnet is based on an ensemble of neural-network-generated devices and directly estimates the gradient of a user's figure of merit to update the weights of a neural network, so that its generated devices are globally optimized and become similar to one another. A residual network-based GLOnet (Res-GLOnet) and a Gaussian mixture model-based GLOnet (GMM-GLOnet) are proposed to improve the global optimization capability of GLOnet and demonstrate the inverse design of non-Hermitian meta-gratings and multilayer thin films. The Res-GLOnet and GMM-GLOnet show better global optimization capability than GLOnet and the conventional local optimization."}] Searching for the neural adjoint method mentioned as the basis extended by ENA. arXiv_search(query="neural adjoint optical multilayer thin films inverse design", max_results=10, sort_by="relevance") to=arXiv_search դարձել code 200 [{"arxiv_id":"(Kim et al., 10 Jul 2025)","title":"Interpretable inverse design of optical multilayer thin films based on extended neural adjoint and regression activation mapping","authors":["Nayoung Ahn","Jungho Lee","Gwanho Kim","Yu-Jin Heo","Hera Kim"],"abstract":"We propose an extended neural adjoint (ENA) framework, which meets six key criteria for artificial intelligence-assisted inverse design of optical multilayer thin films (OMTs): accuracy, efficiency, diversity, scalability, flexibility, and interpretability. To enhance the scalability of the existing neural adjoint method, we present a novel forward neural network architecture for OMTs and introduce a material loss function into the existing neural adjoint loss function, facilitating the exploration of material configurations of OMTs. Furthermore, we present the detailed formulation of the regression activation mapping for the presented forward neural network architecture (F-RAM), a feature visualization method aimed at improving interpretability. We validated the efficacy of the material loss by conducting an ablation study, where each component of the loss function is systematically removed and evaluated. The results indicated that the inclusion of the material loss significantly improves accuracy and diversity. To substantiate the performance of the ENA-based inverse design, we compared it against the residual network-based global optimization network (Res-GLOnet). The ENA yielded the OMT solutions of an inverse design with higher accuracy and better diversity compared to the Res-GLOnet. To demonstrate the interpretability, we applied F-RAM to diverse OMT structures with similar optical properties, obtained by the proposed ENA method. We showed that distributions of feature importance for various OMT structures exhibiting analogous optical properties are consistent, despite variations in material configurations, layer number, and thicknesses. Furthermore, we demonstrate the flexibility of the ENA method by restricting the initial layer of OMTs to SiO2 and 100 nm."},{"arxiv_id":"(2207.13780)","title":"Neural-adjoint method for the rapid design of dielectric metasurfaces","authors":["Yubin Deng","Hui Ge","Zheng Zhang"],"abstract":"The inverse design of nanophotonic metastructures remains challenging due to high computational costs associated with direct simulation. Here, a bidirectional tandem neural network model, named neural-adjoint method, is presented for the inverse design of all-dielectric metasurfaces. Compared to traditional tandem neural network models, the proposed neural-adjoint model is easy to train and possesses less restrictive requirements on the uniqueness of data mappings. The physical significance and equivalence of the proposed neural-adjoint model and the conventional adjoint methods are elaborated. Numerical examples reveal that the proposed model can achieve remarkable accuracy and significantly reduce offline and online computational costs with a slight compromise on design diversity."}] Extended Neural Adjoint (ENA) is a framework for artificial intelligence-assisted inverse design of optical multilayer thin films (OMTs) that combines a scalable forward surrogate, a differentiable objective over thicknesses and material identities, and a regression activation mapping for interpretability. In the formulation reported by Ahn, Lee, Kim, Heo, and Kim, ENA is designed to meet six criteria for AI-assisted inverse design of OMTs: accuracy, efficiency, diversity, scalability, flexibility, and interpretability. Its central components are the OMT forward neural network (OMT-FNN), a material-aware extension of the neural adjoint loss, and F-RAM, a feature-visualization method that assigns per-layer importance to the predicted spectrum (Kim et al., 10 Jul 2025).

1. Problem formulation and design space

The inverse problem is to identify an OMT stack whose optical response matches a target spectrum over a wavelength grid. In the reported setting, the forward outputs are normally the transmittance spectrum $T(\lambda)$ sampled at 301 wavelengths from 400 to 1000 nm at normal incidence, although the same framework also accommodates reflectance $R(\lambda)$, oblique incidence, and s/p polarizations. A stack is parameterized by a layer count $N$, per-layer thicknesses $d=\{d_1,\ldots,d_N\}$, and per-layer material identities $m=\{m_1,\ldots,m_N\}$, with $N \in \{4,8,12,16,20,24\}$ and $d_i \in [20,100]$ nm. Materials are selected from a library of 11 dielectrics with wavelength-dependent refractive indices: MgF2, SiO2, MgO, Al2O3, HfO2, ZnO, Si3N4, AlN, ZnS, TiO2, and ZnSe. Adjacent layers may not repeat the same material, and hard constraints such as fixing the first layer to SiO2 with 100 nm thickness are supported (Kim et al., 10 Jul 2025).

Variable-length stacks are handled by fixing a maximum length $d_i=24$ and padding shorter stacks with a sentinel value $-1$. Padding is neutralized by masking so that only the first $N$ rows contribute to features and loss. Materials are represented as integer indices, but during inverse design ENA treats material variables as continuous relaxations within a valid material range and rounds them to integers only after optimization. The paper states that this relaxation obviates discrete sampling tricks such as Gumbel-softmax while retaining gradient-based efficiency; stability is enforced by material losses and by a final feasibility filter. This design makes the search space explicitly mixed continuous/discrete while keeping the optimization differentiable.

2. Forward surrogate and simulation basis

The OMT-FNN is a light-weight sequence model that predicts the spectrum from a variable-length sequence of material-thickness pairs. It comprises three modules. OMT embedding (OMT-E) receives thickness $R(\lambda)$0 and material index $R(\lambda)$1, both min-max normalized, and uses a 1D convolution to fuse them into an embedding $R(\lambda)$2. OMT feature extraction (OMT-FE) then mixes information along the layer dimension and embedding dimension with two MLP blocks, layer normalization, GELU, masking of padded rows, and a residual skip. If $R(\lambda)$3, the reported operations are

$R(\lambda)$4

with $R(\lambda)$5 and $R(\lambda)$6, followed by

$R(\lambda)$7

with $R(\lambda)$8 and $R(\lambda)$9. The resulting stack feature matrix is $N$0. OMT regression (OMT-R) aggregates over the layer axis with an OMT-wise mean and projects to the spectrum through

$N$1

where $N$2 and $N$3 for $N$4 (Kim et al., 10 Jul 2025).

Training data are generated with a standard transfer-matrix method (TMM) simulator that cascades $N$5 characteristic matrices. For TE/s-polarized light, the interface Fresnel coefficients between layers $N$6 and $N$7 are reported as

$N$8

with Snell’s law $N$9. Inside layer $d=\{d_1,\ldots,d_N\}$0, the phase is

$d=\{d_1,\ldots,d_N\}$1

and the characteristic matrix is

$d=\{d_1,\ldots,d_N\}$2

with $d=\{d_1,\ldots,d_N\}$3 for TE and $d=\{d_1,\ldots,d_N\}$4 for TM. The total transfer matrix is $d=\{d_1,\ldots,d_N\}$5, and the resulting reflectance and transmittance are $d=\{d_1,\ldots,d_N\}$6 and $d=\{d_1,\ldots,d_N\}$7.

The normal-incidence dataset comprises 4,375,000 OMTs with $d=\{d_1,\ldots,d_N\}$8, no adjacent material repetition, thicknesses in $d=\{d_1,\ldots,d_N\}$9 nm at 1 nm resolution, and spectra sampled at 301 points from 400 to 1000 nm. Additional 437,500 OMTs per angle/polarization case support transfer learning for s and p polarizations at 20°, 40°, and 60°. With an 8:1:1 train/validation/test split and Adam at learning rate $m=\{m_1,\ldots,m_N\}$0, the OMT-FNN attains RMSE = 0.010 and $m=\{m_1,\ldots,m_N\}$1 on the held-out test set, with a training time of 33.2 hours on an NVIDIA RTX 4090. Compared to CNN, transformer encoder, and plain MLP feature extractors, the OMT-FE is reported to achieve transformer-level accuracy with about half the parameters and substantially lower training time; transfer learning on a dataset only 1/10 the original size maintains $m=\{m_1,\ldots,m_N\}$2 across the oblique-incidence s/p cases.

3. Material-aware neural adjoint objective

ENA performs inverse design by backpropagating through the frozen OMT-FNN and directly updating design variables. For a target spectrum $m=\{m_1,\ldots,m_N\}$3 and a batch of $m=\{m_1,\ldots,m_N\}$4 candidate designs indexed by $m=\{m_1,\ldots,m_N\}$5, the baseline neural adjoint design loss is

$m=\{m_1,\ldots,m_N\}$6

Thickness feasibility is enforced with a boundary loss

$m=\{m_1,\ldots,m_N\}$7

where $m=\{m_1,\ldots,m_N\}$8 and $m=\{m_1,\ldots,m_N\}$9 denote the mean and magnitude of the valid thickness range. The extension beyond thickness-only optimization is the material loss

$N \in \{4,8,12,16,20,24\}$0

with

$N \in \{4,8,12,16,20,24\}$1

and a material redundancy regularizer $N \in \{4,8,12,16,20,24\}$2 that penalizes adjacent equality. In the reported implementation, material indices are optimized as continuous variables within bounds, and $N \in \{4,8,12,16,20,24\}$3 steers $N \in \{4,8,12,16,20,24\}$4 away from repeated neighboring values. Constraint handling is introduced through

$N \in \{4,8,12,16,20,24\}$5

where $N \in \{4,8,12,16,20,24\}$6 indexes constrained entries. The total loss is

$N \in \{4,8,12,16,20,24\}$7

with masking to exclude padded rows from loss contributions (Kim et al., 10 Jul 2025).

The optimization procedure initializes a population of $N \in \{4,8,12,16,20,24\}$8 candidates with diverse layer counts drawn uniformly from $N \in \{4,8,12,16,20,24\}$9, random thicknesses and materials within valid ranges, and padding to $d_i \in [20,100]$0 with $d_i \in [20,100]$1. At each iteration, the frozen surrogate predicts spectra, the losses are evaluated, and the continuous relaxations $d_i \in [20,100]$2 are updated by gradient descent with Adam at learning rate $d_i \in [20,100]$3:

$d_i \in [20,100]$4

After convergence, optimized material vectors are rounded to integers, and candidates violating bounds or adjacent non-repetition are filtered. The procedure does not require softmax or Gumbel-softmax. According to the paper, the material boundary and redundancy losses provide stable differentiable guidance, while multi-start populations across layer counts and random initializations support diverse solutions.

4. Interpretability through F-RAM

F-RAM is the interpretability component of ENA and adapts regression activation mapping to the OMT-FNN architecture. For a given OMT, the feature matrix $d_i \in [20,100]$5 is obtained by applying OMT-E, OMT-FE, and layer normalization. The final linear layer of OMT-R maps these features to the spectrum, and the reported F-RAM matrix is

$d_i \in [20,100]$6

Its entries are described as inner products between layer features and output weights, with magnitudes reflecting contribution strength. The per-layer importance distribution is then obtained by an embedding-wise mean of absolute entries followed by normalization:

$d_i \in [20,100]$7

Masking ensures that padding rows have negligible importance (Kim et al., 10 Jul 2025).

The reported qualitative role of F-RAM is twofold. First, it reveals that OMTs with similar spectra, even when they differ in materials, thicknesses, and layer counts, induce nearly identical importance profiles. Second, it provides a temporal view of ENA optimization: most structural change occurs in early iterations, while the importance pattern converges quickly and later iterations mainly fine-tune thickness and material values. This suggests that the surrogate has learned invariances at the level of layerwise contribution patterns rather than only at the level of exact material sequences.

5. Empirical behavior: ablation, benchmark, and constrained design

An ablation study isolates the contribution of the material loss using a fixed test target and a 1,200-population run, with 200 candidates per layer count and $d_i \in [20,100]$8. Three variants are compared: ENA without material loss ($d_i \in [20,100]$9), ENA without the material redundancy regularizer $d_i=24$0, and full ENA. The full model achieves RMSE 0.020 and $d_i=24$1 0.992. Removing the material loss degrades performance to RMSE 0.027 and $d_i=24$2 0.985, while removing only $d_i=24$3 degrades it further to RMSE 0.041 and $d_i=24$4 0.963. Diversity is also most favorable for full ENA: removing $d_i=24$5 leads to 586 filtered candidates for adjacent material repetition, versus 78 for full ENA. Relative to ENA without material loss, full ENA yields 82 additional selected solutions; relative to ENA without $d_i=24$6, it increases the selected count by 84. The paper attributes these gains to exploration of material configurations rather than thickness permutations, and states that all three variants complete in seconds to minutes on a single GPU, with negligible overhead from $d_i=24$7 relative to forward prediction cost (Kim et al., 10 Jul 2025).

The principal external benchmark is the residual network-based global optimization network, Res-GLOnet (Polášková et al., 2021). For the band-pass target defined by $d_i=24$8 for 600–700 nm and $d_i=24$9 elsewhere, Res-GLOnet is reimplemented and run at fixed layer counts $-1$0 with 600 population and 2000 epochs sequentially per $-1$1, whereas ENA is run once with total population 3,600, corresponding to 600 per layer count, in parallel across $-1$2 with $-1$3, $-1$4, and $-1$5. At $-1$6, ENA reports RMSE 0.111 and $-1$7 0.912, while Res-GLOnet reports RMSE 0.114 and $-1$8 0.907. Res-GLOnet returns 27 selected OMTs at $-1$9; ENA returns 7 at $N$0 and 2 at $N$1 that also meet the target criterion. The crucial distinction reported in the paper is dispersion: ENA’s average standard deviations in thickness and material index are 7.4× and 50.2× larger than those of Res-GLOnet, and the runtime to produce $N$2 solutions is 131.2 s for ENA versus 6,789.5 s for Res-GLOnet.

Flexibility is demonstrated by adding a hard design-side condition through the constraint loss or by fixing variables directly. Imposing “first layer is SiO2 with 100 nm” with $N$3 across $N$4 yields best RMSEs from 0.033 to 0.039 with $N$5 from 0.967 to 0.977. Among 151 selected solutions, 127 satisfy the first-layer condition as SiO2 near 100 nm, while the remaining cases are filtered or penalized. Even under this restriction, the total selected-solution count remains higher than in the ablations without material exploration.

6. Worked example, limitations, and broader applicability

For the band-pass target $N$6 in 600–700 nm and $N$7 outside, one reported 20-layer ENA solution uses the material sequence MgO / Al2O3 / ZnO / MgF2 / TiO2 / ZnSe / MgF2 / ZnO / Si3N4 / ZnO / ZnSe / ZnO / MgO / Al2O3 / SiO2 / Al2O3 / MgO / ZnS / TiO2 / Al2O3 with thicknesses

30.9 / 31.0 / 64.0 / 91.0 / 20.2 / 24.8 / 75.3 / 55.0 / 20.1 / 27.9 / 76.4 / 99.3 / 35.9 / 21.1 / 78.4 / 63.7 / 62.9 / 85.3 / 55.2 / 67.6 nm. This design achieves RMSE 0.020 and $N$8 0.992. Its F-RAM importance distribution is reported to closely match that of the target structure despite different materials and thicknesses, and during optimization the layer-importance profile converges within approximately 50 iterations before later iterations polish residual errors (Kim et al., 10 Jul 2025).

The reported limitations are tied to the surrogate-training regime and to the continuous relaxation of discrete materials. The forward model is trained on a finite curated subset of an astronomically large design space, estimated as about $N$9 of potential configurations, and assumes no adjacent material repetition. Extreme extrapolation outside the training ranges in layer count, thickness bounds, or material set may therefore degrade accuracy. Although continuous material indices work effectively in the reported setting, the final rounding step can induce discontinuities when optimization terminates near ambiguous boundaries; the boundary and redundancy losses mitigate but do not eliminate this issue. Performance is explicitly sensitive to surrogate accuracy, support for oblique incidence and polarization depends on transfer learning datasets rather than arbitrary-angle generalization, and fabrication tolerances are not modeled in the loss.

The authors present ENA as a general recipe beyond OMTs: train an accurate variable-length surrogate, define differentiable objectives over mixed continuous and discrete parameters, backpropagate to designs, then round and filter. A plausible implication is that the method is especially compatible with inverse problems in which structural length varies and categorical material choices are central. In the paper’s discussion of broader applicability, metasurfaces and multilayer metastructures are cited as natural extensions, with the OMT-FNN replaced by a surrogate over the relevant geometric and material descriptors and F-RAM reused to expose which spatial or geometric components dominate a spectral response.