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Machine Learning-Assisted Analysis and Inverse Design of Prism-Based Surface Plasmon Resonance Sensors

This paper presents a data-driven machine learning framework that utilizes physics-based surrogate models and explainable AI to achieve rapid, high-accuracy inverse design and optimization of Kretschmann-configuration SPR sensors, reducing computational costs by orders of magnitude compared to traditional transfer matrix simulations.

Original authors: R. Runthala, S. Murai, P. Arora

Published 2026-08-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: R. Runthala, S. Murai, P. Arora

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Optical sensors that can detect the tiniest traces of viruses, bacteria, or chemicals without needing to label them with fluorescent dyes have long been a goal for scientists and doctors. These devices rely on a phenomenon called surface plasmon resonance, where light hitting a thin layer of metal causes electrons to ripple in a way that is incredibly sensitive to whatever is touching the surface. To build a sensor that works well, engineers must stack different layers of materials—glass, metal, and special coatings—on top of each other. The performance of the final device depends entirely on the precise thickness of each layer and the specific optical properties of the materials chosen. However, finding the perfect combination is like searching for a needle in a haystack that keeps getting bigger; every time a new material is added or a layer is made slightly thicker, the number of possible designs explodes. Traditionally, testing these designs required running complex computer simulations for each variation, a process that could take seconds per attempt but would still take years to explore all the possibilities.

A team of researchers has now shown how to bypass this slow, tedious search by teaching computers to predict the best designs almost instantly. Instead of running the heavy simulations repeatedly, the team first generated a massive library of over 150,000 simulated sensor designs using a standard physics method. They then trained several different types of machine learning models on this data, teaching them to recognize the patterns that link a sensor's physical structure to its performance. The goal was to find a model that could act as a fast, accurate stand-in for the slow physics simulations, allowing engineers to instantly know how a new design would behave. The researchers tested four distinct machine learning approaches, comparing how well each one could predict two critical measures of success: how sharp the sensor's signal is and how deep the signal dips when it detects a target.

The study found that while all the machine learning models learned the task well, the most successful ones were based on a technique called gradient boosting, which builds a prediction by combining many simple decision trees. These models achieved an accuracy so high that their predictions matched the slow, traditional physics simulations almost perfectly, with errors often less than one percent. More importantly, these trained models could make a prediction in a fraction of a millisecond, whereas the traditional method took several seconds. This represents a speedup of thousands of times, turning a process that would have taken months of computing time into something that happens in the blink of an eye.

Beyond just predicting performance, the researchers used these fast models to work backward, a process known as inverse design. Instead of asking, "If I build this, what will it do?", they asked, "What should I build to get this specific result?" They fed the machine learning models a target performance goal and let optimization algorithms search for the exact combination of materials and thicknesses needed to achieve it. The results were striking: when different optimization algorithms were given the same task, they independently converged on nearly identical sensor designs. To ensure these designs were real and not just mathematical artifacts, the researchers took the top designs found by the machine learning and ran them through the original, slow physics simulations. The real-world physics confirmed the machine's predictions, with the actual performance matching the predicted results within a tiny margin of error.

The study also looked deeply into why the models made the decisions they did, using a method that breaks down the importance of each input. This analysis revealed that the most critical factor for a sensor's performance was the light-absorbing quality of the metal layer and its thickness. While other layers, such as the glass prism or the dielectric coatings, played a role, they were secondary to the metal's ability to dampen or support the electron ripples. By understanding these relationships, the machine learning models did not just guess; they learned the underlying physics of the system. The researchers also tested how robust these models were by slightly shaking the input numbers to simulate the small errors that happen during real-world manufacturing. Even with these realistic variations, the best models remained stable and reliable, suggesting they could be trusted for actual device fabrication.

This work demonstrates that the bottleneck in designing advanced optical sensors is no longer the lack of computing power to simulate physics, but the inefficiency of using those simulations for every single design choice. By replacing the slow, repetitive calculations with a fast, trained machine learning model, engineers can now explore a vast universe of sensor designs in minutes rather than years. The study confirms that this approach is not only faster but also physically accurate, providing a reliable roadmap for creating highly sensitive sensors for medical diagnostics, environmental monitoring, and security. The findings suggest that the future of sensor design lies in this partnership between human intuition and machine speed, where the computer handles the exhaustive search, leaving the human to focus on the most promising discoveries.

Technical Summary: Machine Learning-Assisted Analysis and Inverse Design of Prism-Based Surface Plasmon Resonance Sensors

Problem Statement
Surface Plasmon Resonance (SPR) sensors are critical for label-free, high-sensitivity applications in biosensing and medical diagnostics. However, the design of high-performance multilayer SPR sensors is computationally intensive. Traditional design relies on electromagnetic simulations, such as the Transfer Matrix Method (TMM), to evaluate candidate configurations. As the design space expands with the addition of functional layers (e.g., dielectrics and 2D nanomaterials) and material permutations, the number of feasible configurations grows multiplicatively. Exhaustive TMM-based optimization becomes impractical due to the computational burden, where a single simulation takes several seconds, making the exploration of large parameter spaces time-consuming. Furthermore, existing literature often addresses only isolated aspects of the design process (e.g., forward prediction or optimization) without integrating comprehensive frameworks that include surrogate modeling, inverse design, explainability, and robustness assessment.

Methodology
The authors propose a unified, data-driven framework that integrates machine learning (ML) surrogate modeling with global optimization algorithms to accelerate SPR sensor analysis and inverse design.

Dataset Generation: A diverse physics-based dataset of approximately $1.5 \times 10^5$ unique sensor configurations was generated using the TMM in MATLAB. The study focused on the Kretschmann configuration operating at 1550 nm. The dataset encompassed a generalized multilayer structure (Prism/Metal/Dielectric/2D Material/Sensing Medium) with varied materials (BK7, CaF₂, Au, Ag, Al, SiO₂, TiO₂, Si, Antimonene, Fluorinated Graphene, MoS₂) and thicknesses.
Surrogate Modeling: Four state-of-the-art regression models were trained to predict key performance metrics: Figure of Merit (FOM) and Minimum Reflectance ( $R_{min}$ ). The models included Multi-Layer Perceptron (MLP), XGBoost, LightGBM, and CatBoost. The input features were optical properties and layer thicknesses; the outputs were SPR performance parameters.
Model Evaluation & Explainability: Models were evaluated using $R^2$ , RMSE, MAE, and joint metrics (accuracy, precision, recall) for identifying high-performance configurations. SHAP (SHapley Additive exPlanations) analysis was employed to quantify feature importance and interpret model predictions.
Robustness Assessment: A SHAP-weighted perturbation framework was developed to simulate fabrication-induced uncertainties. Gaussian noise was applied to input parameters, weighted by their feature importance, to assess model stability under realistic variations.
Inverse Design & Optimization: Four global optimization algorithms—Differential Evolution (DE), Dual Annealing (DA), DIRECT, and Bayesian Gaussian Processes (BGP)—were integrated with the trained surrogate models to perform inverse design. The goal was to identify configurations maximizing FOM while minimizing $R_{min}$ .
Validation: The framework was validated through parameter recovery (comparing optimized parameters to target designs) and a complementary forward design task. In the latter, independent optimizer runs were cross-validated against direct TMM simulations to confirm physical consistency.

Key Results

Predictive Performance: The ML models achieved exceptional accuracy. CatBoost yielded the highest performance for FOM prediction ( $R^2 > 0.999$ , RMSE $\approx$ 4.48), while LightGBM performed best for $R_{min}$ ( $R^2 \approx 0.996$ , RMSE $\approx$ 0.058). Ensemble boosting methods (CatBoost, LightGBM, XGBoost) consistently outperformed the MLP baseline, particularly in predicting $R_{min}$ .
Feature Importance: SHAP analysis identified the imaginary part of the metal refractive index ( $M_{im}$ ) and metal thickness ( $t_{Mt}$ ) as the dominant factors influencing sensor performance, accounting for the majority of prediction variance. This aligns with established SPR physics regarding plasmon damping and resonance depth.
Robustness: Under SHAP-weighted perturbations simulating fabrication tolerances, CatBoost and LightGBM demonstrated superior robustness, maintaining high $R^2$ values compared to XGBoost and MLP.
Optimization Efficiency: The surrogate-assisted optimization reduced computational time from seconds per configuration (TMM) to milliseconds, representing a $10^3$ – $10^4$ -fold acceleration.
Inverse Design Success: In the inverse design task, Differential Evolution (DE) and Dual Annealing (DA) achieved the best parameter and performance recovery, with average errors around 1% for FOM and $R_{min}$ .
Forward Design Consensus: In a forward design task where optimizers independently searched for the best configuration without predefined targets, three of the four algorithms (DE, DA, BGP) converged to the exact same optimal sensor configuration. Direct TMM validation of this configuration confirmed the ML predictions with errors of approximately 0.6–0.9% for FOM and under 0.7% for $R_{min}$ .

Significance and Claims
The paper claims that integrating surrogate ML models with optimization algorithms provides a highly effective, computationally efficient framework for rapid SPR sensor analysis and inverse design. The primary significance lies in the comprehensive nature of the workflow, which unifies:

Surrogate Modeling: Achieving high-fidelity predictions ( $R^2 > 0.99$ ) that drastically reduce computational costs.
Explainability: Using SHAP to validate that the ML models learn physically meaningful relationships rather than statistical artifacts.
Robustness: Demonstrating that the models remain reliable under realistic input perturbations.
Inverse Design Validation: Proving that the optimized configurations are not only mathematically optimal but also physically realizable, as confirmed by independent TMM simulations and cross-optimizer consensus.

The authors modestly conclude that while specific models (CatBoost, LightGBM) and optimizers (DE, DA) showed superior performance in this study, the work primarily demonstrates the feasibility and effectiveness of the ML-assisted approach. They note that future refinements could address limitations such as the coarse discretization of 2D material parameters to further enhance generalization across broader design spaces.

Technical Summary: Machine Learning-Assisted Analysis and Inverse Design of Prism-Based Surface Plasmon Resonance Sensors

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