Source-linked AI summary
PRISM: Position-encoded Regressive Inverse Spectral Model for Multilayer Thin-Film Design
Runtian Wang, Renhao Xue, Baige Chen, Hao Wu
TL;DR
Multilayer thin-film inverse design requires selecting materials and thicknesses in a difficult mixed discrete-continuous optimization problem. PRISM uses a unified decoder-only autoregressive transformer with spectrum prefix conditioning and cumulative-depth RoPE, achieving strong accuracy and faster inference than classical optimization. Its reported results include state-of-the-art in-distribution performance and robust scaling and generalization across evaluated conditions.
Problem
Inverse multilayer thin-film design must find material-and-thickness stacks whose computed spectra match target spectra in a combinatorial-continuous optimization problem.
Method
PRISM is a unified decoder-only autoregressive transformer that combines spectrum prefix conditioning, cumulative-depth RoPE, and joint material prediction with continuous thickness regression.
Results
A 44M-parameter PRISM model achieves greedy MAE = 0.012 on in-distribution data, surpasses all baselines including simulated annealing, and runs significantly faster.
Takeaways & Limitations
PRISM provides an efficient alternative to iterative inverse-design methods while maintaining strong accuracy on in-distribution and practical out-of-distribution targets.
Takeaways & Limitations
PRISM requires a full target spectrum at normal incidence and does not accept angle of incidence, excluding partial-spectrum and angular-selectivity designs.
Abstract
from arXiv · showhide
The inverse problem of multilayer thin-film optical coatings design represents a complex combinatorial-continuous optimization challenge. We present PRISM (Position-encoded Regressive Inverse Spectral Model), a unified decoder-only autoregressive transformer that streamlines this process by jointly predicting discrete material selection and continuous thickness regression within a single backbone. PRISM introduces two primary architectural innovations: (1) spectrum prefix conditioning, which utilizes standard prefix tokens for in-context target injection, and (2) cumulative-depth Rotary Position Embeddings, which encode continuous thickness directly into the positional representation to preserve the physical spatial relationships of the stack. Our benchmarks demonstrate that a PRISM-13M model reduces MAE by over 50\% compared to other transformer baselines while utilizing only one-fifth of the parameters. Furthermore, a 44M-parameter variant achieves state-of-the-art performance (MAE = 0.010) on our in-distribution validation benchmark and operates significantly faster than simulated annealing, offering a highly efficient alternative to classical optimization methods.
1. Introduction
PRISM targets the mixed discrete-continuous inverse design of multilayer optical coatings with a unified autoregressive transformer. Its spectrum conditioning and physical depth encoding support efficient joint material and thickness prediction, with strong benchmark performance.
- The inverse coating-design problem jointly selects materials and continuous layer thicknesses in a highly non-convex, degenerate search space.
- Traditional methods evaluate thousands of candidate structures through TMM, requiring minutes to hours per design.
- PRISM combines factored material prediction and continuous thickness regression within a single decoder-only backbone.
- Spectrum prefix conditioning prepends a projected target-spectrum token, keeping conditioning visible under causal self-attention.
- Cumulative-depth RoPE uses physical stack depth in nanometers instead of sequential token indices to encode geometric relationships.
- PRISM substantially outperforms neural baselines and simulated annealing on in-distribution benchmarks while remaining competitive on practical targets with faster inference.
2. Related Work
Prior inverse-design methods include iterative numerical optimization, fixed-length neural representations, conditional generative models, and autoregressive transformers. PRISM’s related-work position is to avoid joint material-thickness vocabularies without adopting a dual-decoder design.
- Classical inverse thin-film design: Needle optimization, genetic algorithms, and simulated annealing search the joint discrete-continuous design space using iterative or stochastic procedures.
- Neural inverse design in nanophotonics: Tandem networks use fixed-length representations, limiting their ability to handle variable-length coating designs naturally.
- Neural inverse design in nanophotonics: Conditional GANs and CVAEs model distributions conditioned on target spectra but often struggle with mode coverage and require multiple samples.
- Autoregressive models for inverse design: OptoGPT uses a joint material-discretized-thickness vocabulary that scales as |materials| × |thickness bins|, reaching 904 tokens for 18 materials and 50 bins.
- Concurrent work: OptoFormer separates material and thickness generation with two decoders, avoiding vocabulary explosion but increasing architectural complexity.
- Rotary Position Embeddings: PRISM repurposes RoPE by using cumulative physical depth in nanometers as position for this non-sequential design domain.
3. Method
PRISM formulates thin-film inverse design as autoregressive generation of material choices and continuous thicknesses. Its decoder-only architecture conditions on the target spectrum, encodes cumulative physical depth, and uses shared material and thickness heads for decoding and beam search.
- Problem formulation: A thin-film stack is represented by material-thickness pairs, and the inverse problem seeks a design whose TMM-computed spectrum matches a target spectrum.The spectrum concatenates sampled reflectance and transmittance values.
- Autoregressive formulation: PRISM generates layers left to right, selecting among 17 materials and regressing a continuous thickness until emitting EOS.Material selection is categorical, while thickness is a continuous regression target.
- Spectrum prefix conditioning: The decoder-only model prepends a learned linear projection of the target spectrum and uses causal attention so later tokens can always access the conditioning token.This replaces an encoder and cross-attention with conditioning inside the causal attention window.
- Cumulative-depth RoPE: Cumulative-depth RoPE assigns each token the running sum of layer thicknesses rather than an integer layer index, encoding physical separation in nanometers.The spectrum prefix is assigned position p0 = 0, and continuous positions are applied to queries and keys.
- Dual output heads: A shared transformer backbone feeds a material-logit head and a per-material thickness regression head, with thickness outputs indexed by the selected material.The per-material head avoids a two-stage decode when thickness depends on the chosen material.
- Inference: During beam search, PRISM re-simulates candidates with TMM and reranks them by true spectral error against the target.The paper states that this adds negligible cost because TMM evaluation is cheap.
4. Experimental Setup
PRISM is evaluated across a 17-material thin-film design space using TMM-generated data, two model scales, five baselines, and in-distribution and practical-target benchmarks.
- Data and design space: The design space contains 17 materials, 1–20 layers per stack, and thicknesses from 10 to 500 nm in 10 nm steps.Spectra contain reflectance and transmittance at 71 wavelengths from 400–1100 nm.
- Data and design space: Training uses uniformly sampled material sequences and thicknesses simulated with TMM, with up to 30M training, 100K development, and 10K validation samples.Longer sequences are oversampled using P(L) ∝ L.
- Models and training: PRISM is trained at 13M and 44M parameters to study scale, with the smaller model matched to OptoGPT’s data conditions and the larger model trained for maximum performance.The models use 30 epochs on 10M samples and 60 epochs on 30M samples, respectively.
- Baselines: The comparison includes simulated annealing, differentiable TMM, OptoGPT, Tandem Network, and CVAE on a common validation set and practical targets.Optimization baselines use iterative restarts, while neural baselines include autoregressive, inverse-forward, and conditional-generative approaches.
- Evaluation: Performance is measured on TMM-resimulated 142-dimensional spectra using MAE and R2, with EMD additionally reported for practical targets.PRISM is evaluated with greedy decoding and TMM-reranked beam search.
- Evaluation: The benchmarks comprise 10,000 generated targets matching training and 84 practical filter spectra absent from the training distribution.The practical set spans narrowband, broadband, edge, notch, dichroic, mirror, beam-splitter, and other filter categories.
5. Results
PRISM outperforms neural and optimization baselines on the in-distribution validation benchmark, while practical-target results depend on whether pointwise or shape-sensitive error is measured. Scaling and TMM reranking further improve performance across evaluations.
- In-distribution performance: PRISM-44M greedy outperforms all methods, including simulated annealing, on the in-distribution validation set.PRISM-13M also outperforms all neural baselines despite having 4.9× fewer parameters than OptoGPT.
- Practical targets: On practical targets, PRISM substantially outperforms all prior neural baselines on every reported metric.The evaluation uses TMM-resimulated spectra and all 84 practical targets.
- Out-of-distribution sequence lengths: Both PRISM-13M and PRISM-44M maintain robust greedy MAE when extrapolating to sequences up to 2.5× longer than training.Scaling from 13M to 44M parameters halves greedy MAE across the tested conditions.
6. Analysis
Analysis shows that practical-target rankings depend on metric choice, while PRISM generalizes to substantially longer generated sequences with limited degradation.
- Spectral shape fidelity: Neural methods, especially PRISM, track practical target shapes more faithfully than optimization baselines in qualitative spectral comparisons.They capture sharper band edges, peak positions, and passband widths that pointwise MAE under-credits.
- Sequence-length generalization: PRISM exhibits minimal degradation on generated sequences up to 2.5× longer than training for both model sizes.Figure 2 evaluates 10,000 samples per condition with 10 nm thickness steps.
- Metric interpretation: Pointwise MAE rewards average agreement rather than structural alignment, which can favor smoothed spectra that lose key structures.This explains why simulated annealing’s pointwise advantage does not determine the shape-sensitive ranking.
7. Conclusion
PRISM combines spectrum prefix conditioning with cumulative-depth RoPE for inverse thin-film design, achieving strong in-distribution and practical-target results while revealing important conditioning limits.
- PRISM introduces spectrum prefix conditioning and cumulative-depth RoPE as its two architectural innovations.The former injects the target spectrum, while the latter encodes cumulative physical depth in the positional representation.
- A 44M-parameter PRISM model achieves greedy MAE = 0.012 on in-distribution data and runs significantly faster than simulated annealing.It surpasses all baselines, including simulated annealing, on this benchmark.
- On 84 out-of-distribution practical targets, PRISM-44M TMM-reranked achieves the lowest spectral earth-mover’s distance despite simulated annealing’s lower pointwise MAE.The figure comparison also indicates that PRISM and other neural methods can track target curves more closely than optimization baselines.
- PRISM maintains greedy MAE on sequences up to 2.5× longer than training, while scaling from 13M to 44M parameters halves greedy MAE across conditions.Both model scales exhibit the same qualitative behaviours.
- The current conditioning scheme requires a full target spectrum at normal incidence and does not support partial wavelength specifications or angle-of-incidence inputs.Richer conditioning interfaces are identified as future work.