Active learning for photonics

TL;DR

This work addresses the data-hungry task of predicting photonic crystal band gaps by introducing an analytic last-layer Bayesian neural network (LL-BNN) within an active learning loop. The LL-BNN provides a closed-form predictive variance $s(x)$, enabling uncertainty-driven sample selection that prioritizes the most informative simulations and avoids MC sampling overhead. On a dataset of $11{,}376$ two-tone 2D photonic crystals, the approach achieves up to $2.6\times$ data savings while maintaining accuracy, demonstrating substantial improvements in data efficiency for surrogate modeling in photonics. The framework is general and readily extensible to other scientific regression tasks, offering a scalable path toward faster inverse design and topological optimization in photonics and beyond.

Abstract

Active learning for photonic crystals explores the integration of analytic approximate Bayesian last layer neural networks (LL-BNNs) with uncertainty-driven sample selection to accelerate photonic band gap prediction. We employ an analytic LL-BNN formulation, corresponding to the infinite Monte Carlo sample limit, to obtain uncertainty estimates that are strongly correlated with the true predictive error on unlabeled candidate structures. These uncertainty scores drive an active learning strategy that prioritizes the most informative simulations during training. Applied to the task of predicting band gap sizes in two-dimensional, two-tone photonic crystals, our approach achieves up to a 2.6x reduction in required training data compared to a random sampling baseline while maintaining predictive accuracy. The efficiency gains arise from concentrating computational resources on high uncertainty regions of the design space rather than sampling uniformly. Given the substantial cost of full band structure simulations, especially in three dimensions, this data efficiency enables rapid and scalable surrogate modeling. Our results suggest that analytic LL-BNN based active learning can substantially accelerate topological optimization and inverse design workflows for photonic crystals, and more broadly, offers a general framework for data efficient regression across scientific machine learning domains.

Figures (5)

• Figure 1: Active‐Learning Pipeline for 2D Photonic‐Crystal Band‑Gap Prediction. Starting from the dielectric‐constant map of a 2D two‑tone photonic‑crystal unit cell (augmented via symmetry operations), we feed each sample into a deep neural network whose final layer is approximate Bayesian. In this last layer, each weight and bias is treated as a Gaussian random variable, so a single input yields a predictive distribution rather than a point estimate. When evaluating unlabeled candidates, we compute the output variance as an uncertainty score and select the highest‐variance samples for expensive band‑structure simulations. By iteratively adding only the most informative points, the model rapidly improves its accuracy with far fewer training examples.
• Figure 2: Symmetry-preserving data augmentation for 2D photonic crystal unit cells. Rotations, reflections, and translations of the $32 \times 32$ permittivity maps leave the photonic band structure unchanged. During training, random combinations of these transformations are applied to each sample, effectively enlarging the dataset while enforcing physical invariance of the band gap labels.
• Figure 3: BNN Uncertainty Calibration. After training on 50 randomly selected samples, we evaluate our approximate Bayesian last layer on the held‑out test set. For each test point, we compute the predictive standard deviation $s(x)$ and the squared error $(y - \hat{y})^2$. We then sort all test samples by $s(x)$, partition them into 15 bins of 100 samples each, and plot the mean squared error (MSE) of each bin along the sorted sample index. The clear downward trend shows that higher estimated uncertainty corresponds to larger true errors, confirming a strong monotonic relationship even when the overall model accuracy is low.
• Figure 4: Spearman Coefficient over Active Learning. At each active learning step we compute the Spearman rank correlation on the test dataset. As it consistently remains negative, we see that the model's uncertainty is inversely correlated to true error through the wide range of training dataset sizes. We also note the uncertainty-based sample selection in the Analytic LL-BNN reduces the magnitude of this correlation, since -1 signifies perfect correlation.
• Figure 5: Comparison of Random vs Uncertainty-Driven Sampling. Test set MSE for photonic‐crystal band‑gap prediction as a function of total training samples, averaged over ten runs (each initialized with a different random pool of 50 examples). Blue shows standard random sampling; orange shows uncertainty sampling via our approximate Bayesian last layer. The green curve shows an "oracle" baseline that, at each iteration, greedily selects the 50 unlabeled samples with the highest true regression error—information unavailable in practice. This oracle is not globally optimal but provides an upper bound on what any greedy acquisition based on uncertainty could achieve. Error bars indicate standard error across runs. Uncertainty sampling not only achieves lower error at every budget level but also exhibits far less run‑to‑run variability and reaches the same accuracy as random sampling with roughly one‑third the data.