Interpretable inverse design of optical multilayer thin films based on extended neural adjoint and regression activation mapping

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.

💡 Research Summary

The paper introduces an Extended Neural Adjoint (ENA) framework designed to meet six essential criteria for AI‑assisted inverse design of optical multilayer thin films (OMTs): accuracy, efficiency, diversity, scalability, flexibility, and interpretability. The authors first develop a novel forward neural network architecture (F‑Net) that accepts layer‑wise material identifiers (one‑hot encoded) and continuous thickness values as inputs and directly predicts the full spectral reflectance and transmittance. By sharing weights across layers and employing multi‑scale convolutional blocks with skip connections, the network remains compact and robust even when the number of layers or material choices vary.

To overcome the limitation of conventional Neural Adjoint (NA) methods, which only minimize a spectral L2 loss, the authors augment the loss function with a “material loss.” This term penalizes deviations from a predefined set of admissible materials using a cross‑entropy formulation, thereby steering the optimization toward physically realizable material configurations while still encouraging a broad set of distinct designs. The total loss combines the spectral error, material loss, and a regularization term that promotes smooth thickness variations.

A comprehensive ablation study evaluates three configurations: (a) spectral loss only, (b) material loss only, and (c) the combined loss. Results show that the combined loss reduces the average spectral RMSE by roughly 33 % and more than doubles the number of unique viable designs, confirming that the material loss substantially enhances both accuracy and diversity.

Interpretability is addressed through Regression Activation Mapping (F‑RAM), a gradient‑based visualization that back‑propagates from the final output layer to quantify the contribution of each input parameter (material or thickness) to the predicted spectrum. When applied to multiple OMT structures that share the same target optical response, F‑RAM consistently highlights similar regions of the parameter space, demonstrating that the network’s decision‑making aligns with physical intuition about which layers dominate the optical performance.

Performance benchmarking against the Residual‑GLOnet (Res‑GLOnet) global optimization network shows that ENA achieves a lower average thickness error (0.032 µm vs. 0.045 µm) and a lower spectral RMSE (0.018 vs. 0.027) while requiring about 30 % less computational time over 5,000 design trials. This confirms ENA’s superior accuracy, efficiency, and ability to generate a more diverse set of solutions.

Flexibility is further demonstrated by imposing a hard constraint on the first layer (SiO₂ with a fixed 100 nm thickness). Even under this restriction, ENA successfully discovers multiple designs that meet the target spectrum, illustrating that the framework can accommodate user‑defined material or geometric constraints without sacrificing performance.

In summary, the ENA framework integrates a scalable forward model, a material‑aware loss, and a gradient‑based interpretability tool to deliver high‑quality, diverse, and physically plausible OMT designs. The authors suggest future extensions to three‑dimensional photonic structures, nonlinear materials, and real‑time integration with fabrication processes, positioning ENA as a versatile platform for next‑generation photonic device engineering.