# Smart and Rapid Design of Nanophotonic Structures by an Adaptive and Regularized Deep Neural Network

^{1}

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## Abstract

**:**

## 1. Introduction

**1**.- To the best of our knowledge, this is the first time V is modeled by a DNN model as an optical property. V is crucial for reducing device footprints and having tight on-chip integration.
**2**.- The employment of CNN empowers the algorithm through its unique advantage on recognizing complex patterns and extracting hidden information from images.
**3**.- The use of learning rate scheduling (also known as adaptive learning rate) can effectively smoothen and speed up the convergence of the training process.
**4**.- The use of L2 regularization can effectively reduce overfitting and improve the generalizibility of LRS-RCNN.
**5**.- It has a high-dimensional design parameter (DA) space with over 160 degrees of freedom. A large DA space is a prerequisite for real-world design problems.

## 2. Methods

#### 2.1. DNN Structure and Architecture

#### 2.2. Algorithm Description and Approach

- 1.
- Adaptive learning rate through learning rate scheduling and, thus, the “LRS” in LRS-RCNN. Adaptive learning rate works by dynamically reducing the learning rate when training slows down or a metric hits a plateau and has the power of gaining robustness against gradient noise and inducing a smoother and faster convergence [1,2,48]. While there is a multitude of learning rate schedulers available, Reduce-On-Plateau was selected in this work due to its stable and consistent behavior according to our experiments.
- 2.
- L2 Regularization and, thus, the “R” after the hyphen in LRS-RCNN. When there is a complex model with a large number of features in the dataset, L2 regularization can be used in backpropagation to address the common overfitting issue and boost generalizibility [1,2]. It works by adding a squared penalty term associated with weight parameters (W) to the loss function, as shown in Equation (1), where $\lambda $ controls how much one would like to penalize large weights. The Error term corresponds to MSE in Equation (2).$$Regularized\phantom{\rule{3.33333pt}{0ex}}Loss=Error({x}_{i},{x}^{*})+\lambda \sum _{i}^{N}{W}_{i}^{2}$$

## 3. Results and Discussion

#### 3.1. Nanobeam

**Table 3.**Results of training LRS-RCNN with nanobeam and L3 nanocavity. Tabulated are the performance metrics of the predictor at the test/validation phase and its comparison against prior works. Data partially extracted from Figure 6 and Figure 8. Overlined parameters, such as $\overline{{\u03f5}_{pred}}$ and $\overline{MSE}$, indicate converged values. Bold-faced numbers are best metrics. Prediction time and speedup vs. FDTD, both calculated by averaging the time measured with samples in the validation dataset, refer to the time for characterizing a single structure by LRS-RCNN. N/A means no data available from the cited work.

min ${\mathit{\u03f5}}_{\mathit{p}\mathit{r}\mathit{e}\mathit{d}}$ | $\overline{{\mathit{\u03f5}}_{\mathit{p}\mathit{r}\mathit{e}\mathit{d}}}$ | min $\mathit{MSE}$ | $\overline{\mathit{MSE}}$ | Epochs | Training Time | Prediction Time | Speedup (vs. FDTD) | ||
---|---|---|---|---|---|---|---|---|---|

Nanobeam | Q | 0.000157% | 0.0148% | 0.00734 | 0.00735 | 700 | 9 min | $7\times {10}^{-5}$ s | $5.1\times {10}^{6}$ |

V | 0.00102% | 0.141% | $\mathbf{3}.\mathbf{00}\times {\mathbf{10}}^{-\mathbf{8}}$ | $\mathbf{3}.\mathbf{16}\times {\mathbf{10}}^{-\mathbf{8}}$ | |||||

L3 | Q | 0.00105% | 0.0112% | 0.000360 | 0.000365 | 300 | 5 min | $9\times {10}^{-5}$ s | $3\times {10}^{6}$ |

V | 0.00317% | 0.0158% | 0.00252 | 0.00282 | |||||

Gou et al. [22] | 0.540% | N/A | 0.000300 | N/A | 1000 | 143 s | $\mathbf{3}.\mathbf{6}\times {\mathbf{10}}^{-\mathbf{5}}$ s | $\mathbf{1}.\mathbf{2}\times {\mathbf{10}}^{\mathbf{8}}$ | |

Christensen et al. [14] | 0.470% | 0.530% | N/A | 0.006 | 400 | 3 min | 0.02 s | N/A | |

Asano et al. [10] | 16.0% | 19.0% | N/A | N/A | $2\times {10}^{5}$ | N/A | N/A | N/A | |

Chugh et al. [15] | 1.00% | N/A | N/A | 0.00808 | 4000 | N/A | N/A | N/A | |

Singh et al. [13] | N/A | N/A | 0.00810 | 0.014 | 500 | 45 min | N/A | N/A | |

Chen et al. [23] | 0.700% | 5.00% | N/A | N/A | 110 | N/A | N/A | N/A | |

Wiecha et al. [18] | N/A | 5.78% | N/A | N/A | 100 | 200 min | $3\times {10}^{-3}$ s | ${10}^{5}$ | |

Tahersima et al. [51] | N/A | N/A | N/A | 0.130 | 10,000 | 22 min | N/A | N/A |

#### 3.2. L3 Nanocavity

#### 3.3. On the Importance of Adaptiveness and Regularization

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- LeCun, Y.; Bengio, Y.; Hinton, G. Deep learning. Nature
**2015**, 521, 436–444. [Google Scholar] [CrossRef] [PubMed] - Goodfellow, I.; Bengio, Y.; Courville, A.; Bengio, Y. Deep Learning; MIT Press: Cambridge, MA, USA, 2016; Volume 1. [Google Scholar]
- Hu, G.; Yang, Y.; Yi, D.; Kittler, J.; Christmas, W.; Li, S.Z.; Hospedales, T. When face recognition meets with deep learning: An evaluation of convolutional neural networks for face recognition. In Proceedings of the IEEE International Conference on Computer Vision Workshops, Santiago, Chile, 7–13 December 2015; pp. 142–150. [Google Scholar]
- Yim, J.; Jung, H.; Yoo, B.; Choi, C.; Park, D.; Kim, J. Rotating your face using multi-task deep neural network. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Boston, MA, USA, 7–12 June 2015; pp. 676–684. [Google Scholar]
- Zhang, M.; Zhang, Y.; Zhang, L.; Liu, C.; Khurshid, S. DeepRoad: GAN-based metamorphic testing and input validation framework for autonomous driving systems. In Proceedings of the 2018 33rd IEEE/ACM International Conference on Automated Software Engineering (ASE), Montpellier, France, 3–7 September 2018; pp. 132–142. [Google Scholar]
- Li, P.; Chen, X.; Shen, S. Stereo r-cnn based 3d object detection for autonomous driving. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, Long Beach, CA, USA, 15–20 June 2019; pp. 7644–7652. [Google Scholar]
- Ma, W.; Liu, Z.; Kudyshev, Z.A.; Boltasseva, A.; Cai, W.; Liu, Y. Deep learning for the design of photonic structures. Nat. Photonics
**2020**, 15, 77–90. [Google Scholar] [CrossRef] - Jiang, J.; Chen, M.; Fan, J.A. Deep neural networks for the evaluation and design of photonic devices. Nat. Rev. Mater.
**2020**, 6, 679–700. [Google Scholar] [CrossRef] - Ma, W.; Cheng, F.; Liu, Y. Deep-learning-enabled on-demand design of chiral metamaterials. ACS Nano
**2018**, 12, 6326–6334. [Google Scholar] [CrossRef] [PubMed] - Asano, T.; Noda, S. Optimization of photonic crystal nanocavities based on deep learning. Opt. Express
**2018**, 26, 32704–32717. [Google Scholar] [CrossRef] [PubMed] - Abe, R.; Takeda, T.; Shiratori, R.; Shirakawa, S.; Saito, S.; Baba, T. Optimization of an H0 photonic crystal nanocavity using machine learning. Opt. Lett.
**2020**, 45, 319–322. [Google Scholar] [CrossRef] - Ma, W.; Liu, Y. A data-efficient self-supervised deep learning model for design and characterization of nanophotonic structures. Sci. China Phys. Mech. Astron.
**2020**, 68, 1–8. [Google Scholar] [CrossRef] - Singh, R.; Agarwal, A.; Anthony, B.W. Mapping the design space of photonic topological states via deep learning. Opt. Express
**2020**, 28, 27893–27902. [Google Scholar] [CrossRef] - Christensen, T.; Loh, C.; Picek, S.; Jakobović, D.; Jing, L.; Fisher, S.; Ceperic, V.; Joannopoulos, J.D.; Soljačić, M. Predictive and generative machine learning models for photonic crystals. Nanophotonics
**2020**, 9, 4183–4192. [Google Scholar] [CrossRef] - Chugh, S.; Ghosh, S.; Gulistan, A.; Rahman, B. Machine learning regression approach to the nanophotonic waveguide analyses. J. Light. Technol.
**2019**, 37, 6080–6089. [Google Scholar] [CrossRef] [Green Version] - Asano, T.; Noda, S. Iterative optimization of photonic crystal nanocavity designs by using deep neural networks. Nanophotonics
**2019**, 8, 2243–2256. [Google Scholar] [CrossRef] - Jiang, J.; Fan, J.A. Global optimization of dielectric metasurfaces using a physics-driven neural network. Nano Lett.
**2019**, 19, 5366–5372. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wiecha, P.R.; Muskens, O.L. Deep learning meets nanophotonics: A generalized accurate predictor for near fields and far fields of arbitrary 3D nanostructures. Nano Lett.
**2019**, 20, 329–338. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Liu, D.; Tan, Y.; Khoram, E.; Yu, Z. Training deep neural networks for the inverse design of nanophotonic structures. Acs Photonics
**2018**, 5, 1365–1369. [Google Scholar] [CrossRef] [Green Version] - Li, R.; Gu, X.; Li, K.; Huang, Y.; Li, Z.; Zhang, Z. Deep learning-based modeling of photonic crystal nanocavities. Opt. Mater. Express
**2021**, 11, 2122–2133. [Google Scholar] [CrossRef] - Sajedian, I.; Badloe, T.; Rho, J. Optimisation of color generation from dielectric nanostructures using reinforcement learning. Opt. Express
**2019**, 27, 5874–5883. [Google Scholar] [CrossRef] - Gou, Z.; Wang, C.; Yang, Y.; Han, Z.; Nie, T.; Tian, H. Artificial neural networks applied in fast-designing ultrabroad bandgap elliptical hole dielectric mode photonic crystal nanobeam cavity. Appl. Opt.
**2021**, 60, 8977–8982. [Google Scholar] [CrossRef] - Chen, Y.; Zhu, J.; Xie, Y.; Feng, N.; Liu, Q.H. Smart inverse design of graphene-based photonic metamaterials by an adaptive artificial neural network. Nanoscale
**2019**, 11, 9749–9755. [Google Scholar] [CrossRef] - Ryu, H.Y.; Notomi, M.; Lee, Y.H. High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities. Appl. Phys. Lett.
**2003**, 83, 4294–4296. [Google Scholar] [CrossRef] [Green Version] - Tanaka, Y.; Asano, T.; Noda, S. Design of Photonic Crystal Nanocavity With Q-Factor of 10
^{9}. J. Light. Technol.**2008**, 26, 1532–1539. [Google Scholar] [CrossRef] [Green Version] - Zhang, Z.; Qiu, M. Small-volume waveguide-section high Q microcavities in 2D photonic crystal slabs. Opt. Express
**2004**, 12, 3988–3995. [Google Scholar] [CrossRef] [PubMed] - Lai, Y.; Pirotta, S.; Urbinati, G.; Gerace, D.; Minkov, M.; Savona, V.; Badolato, A.; Galli, M. Genetically designed L3 photonic crystal nanocavities with measured quality factor exceeding one million. Appl. Phys. Lett.
**2014**, 104, 241101. [Google Scholar] [CrossRef] [Green Version] - Taguchi, Y.; Takahashi, Y.; Sato, Y.; Asano, T.; Noda, S. Statistical studies of photonic heterostructure nanocavities with an average Q factor of three million. Opt. Express
**2011**, 19, 11916–11921. [Google Scholar] [CrossRef] [PubMed] - Kuramochi, E.; Notomi, M.; Mitsugi, S.; Shinya, A.; Tanabe, T.; Watanabe, T. Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect. Appl. Phys. Lett.
**2006**, 88, 041112. [Google Scholar] [CrossRef] - Akahane, Y.; Asano, T.; Song, B.S.; Noda, S. High-Q photonic nanocavity in a two-dimensional photonic crystal. Nature
**2003**, 425, 944–947. [Google Scholar] [CrossRef] - Kaniber, M.; Laucht, A.; Neumann, A.; Villas-Bôas, J.; Bichler, M.; Amann, M.C.; Finley, J. Investigation of the nonresonant dot-cavity coupling in two-dimensional photonic crystal nanocavities. Phys. Rev. B
**2008**, 77, 161303. [Google Scholar] [CrossRef] [Green Version] - Shambat, G.; Ellis, B.; Petykiewicz, J.; Mayer, M.A.; Majumdar, A.; Sarmiento, T.; Harris, J.S.; Haller, E.E.; Vuckovic, J. Electrically driven photonic crystal nanocavity devices. IEEE J. Sel. Top. Quantum Electron.
**2012**, 18, 1700–1710. [Google Scholar] [CrossRef] [Green Version] - Ota, Y.; Liu, F.; Katsumi, R.; Watanabe, K.; Wakabayashi, K.; Arakawa, Y.; Iwamoto, S. Photonic crystal nanocavity based on a topological corner state. Optica
**2019**, 6, 786–789. [Google Scholar] [CrossRef] - Altug, H.; Vučković, J. Photonic crystal nanocavity array laser. Opt. Express
**2005**, 13, 8819–8828. [Google Scholar] [CrossRef] [Green Version] - Barth, M.; Kouba, J.; Stingl, J.; Löchel, B.; Benson, O. Modification of visible spontaneous emission with silicon nitride photonic crystal nanocavities. Opt. Express
**2007**, 15, 17231–17240. [Google Scholar] [CrossRef] - Quan, Q.; Deotare, P.B.; Loncar, M. Photonic crystal nanobeam cavity strongly coupled to the feeding waveguide. Appl. Phys. Lett.
**2010**, 96, 203102. [Google Scholar] [CrossRef] [Green Version] - Quan, Q.; Loncar, M. Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities. Opt. Express
**2011**, 19, 18529–18542. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Karnadi, I.; Son, J.; Kim, J.Y.; Jang, H.; Lee, S.; Kim, K.S.; Min, B.; Lee, Y.H. A printed nanobeam laser on a SiO 2/Si substrate for low-threshold continuous-wave operation. Opt. Express
**2014**, 22, 12115–12121. [Google Scholar] [CrossRef] [PubMed] - Fegadolli, W.S.; Kim, S.H.; Postigo, P.A.; Scherer, A. Hybrid single quantum well InP/Si nanobeam lasers for silicon photonics. Opt. Lett.
**2013**, 38, 4656–4658. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kim, S.; Ahn, B.H.; Kim, J.Y.; Jeong, K.Y.; Kim, K.S.; Lee, Y.H. Nanobeam photonic bandedge lasers. Opt. Express
**2011**, 19, 24055–24060. [Google Scholar] [CrossRef] - Jeong, K.Y.; No, Y.S.; Hwang, Y.; Kim, K.S.; Seo, M.K.; Park, H.G.; Lee, Y.H. Electrically driven nanobeam laser. Nat. Commun.
**2013**, 4, 1–6. [Google Scholar] [CrossRef] [Green Version] - Zhang, Y.; Khan, M.; Huang, Y.; Ryou, J.; Deotare, P.; Dupuis, R.; Lončar, M. Photonic crystal nanobeam lasers. Appl. Phys. Lett.
**2010**, 97, 051104. [Google Scholar] [CrossRef] [Green Version] - Deotare, P.B.; McCutcheon, M.W.; Frank, I.W.; Khan, M.; Lončar, M. High quality factor photonic crystal nanobeam cavities. Appl. Phys. Lett.
**2009**, 94, 121106. [Google Scholar] [CrossRef] [Green Version] - Greenspan, H.; Van Ginneken, B.; Summers, R.M. Guest editorial deep learning in medical imaging: Overview and future promise of an exciting new technique. IEEE Trans. Med. Imaging
**2016**, 35, 1153–1159. [Google Scholar] [CrossRef] - Ren, S.; He, K.; Girshick, R.; Sun, J. Faster R-CNN: Towards real-time object detection with region proposal networks. IEEE Trans. Pattern Anal. Mach. Intell.
**2016**, 39, 1137–1149. [Google Scholar] [CrossRef] [Green Version] - Glorot, X.; Bordes, A.; Bengio, Y. Deep sparse rectifier neural networks. In Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, JMLR Workshop and Conference Proceedings, Fort Lauderdale, FL, USA, 11–13 April 2011; pp. 315–323. [Google Scholar]
- Dumoulin, V.; Visin, F. A guide to convolution arithmetic for deep learning. arXiv
**2016**, arXiv:1603.07285. [Google Scholar] - Zeiler, M.D. Adadelta: An adaptive learning rate method. arXiv
**2012**, arXiv:1212.5701. [Google Scholar] - Lumerical. Lumerical FDTD Solutions. Product Serial Number: 75D28D211BA4.
- Li, R. Repository for Data Files and Datasets. 2022. Available online: https://github.com/Arcadianlee/Deep-Learning-Design-Photonic-Crystals.git (accessed on 10 November 2021).
- Tahersima, M.H.; Kojima, K.; Koike-Akino, T.; Jha, D.; Wang, B.; Lin, C.; Parsons, K. Deep neural network inverse design of integrated photonic power splitters. Sci. Rep.
**2019**, 9, 1–9. [Google Scholar] [CrossRef] - Zhou, T.; Tang, M.; Xiang, G.; Xiang, B.; Hark, S.; Martin, M.; Baron, T.; Pan, S.; Park, J.S.; Liu, Z.; et al. Continuous-wave quantum dot photonic crystal lasers grown on on-axis Si (001). Nat. Commun.
**2020**, 11, 1–7. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Liu, X.; Mashooq, K.; Szkopek, T.; Mi, Z. Improving the efficiency of transverse magnetic polarized emission from AlGaN based LEDs by using nanowire photonic crystal. IEEE Photonics J.
**2018**, 10, 1–11. [Google Scholar] [CrossRef] - Feldmann, J.; Youngblood, N.; Wright, C.D.; Bhaskaran, H.; Pernice, W. All-optical spiking neurosynaptic networks with self-learning capabilities. Nature
**2019**, 569, 208–214. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shastri, B.J.; Tait, A.N.; de Lima, T.F.; Pernice, W.H.; Bhaskaran, H.; Wright, C.D.; Prucnal, P.R. Photonics for artificial intelligence and neuromorphic computing. Nat. Photonics
**2021**, 15, 102–114. [Google Scholar] [CrossRef] - Gan, X.; Gao, Y.; Fai Mak, K.; Yao, X.; Shiue, R.J.; Van Der Zande, A.; Trusheim, M.E.; Hatami, F.; Heinz, T.F.; Hone, J.; et al. Controlling the spontaneous emission rate of monolayer MoS2 in a photonic crystal nanocavity. Appl. Phys. Lett.
**2013**, 103, 181119. [Google Scholar] [CrossRef] [Green Version] - Zhang, H.; Zhang, X.; Li, H.; Deng, Y.; Zhang, X.; Xi, L.; Tang, X.; Zhang, W. A design strategy of the circular photonic crystal fiber supporting good quality orbital angular momentum mode transmission. Opt. Commun.
**2017**, 397, 59–66. [Google Scholar] [CrossRef] - Mahmoodian, S.; Prindal-Nielsen, K.; Söllner, I.; Stobbe, S.; Lodahl, P. Engineering chiral light–matter interaction in photonic crystal waveguides with slow light. Opt. Mater. Express
**2017**, 7, 43–51. [Google Scholar] [CrossRef] [Green Version] - Hendrickson, J.R.; Soref, R.; Gibson, R. Improved 2× 2 Mach–Zehnder switching using coupled-resonator photonic-crystal nanobeams. Opt. Lett.
**2018**, 43, 287–290. [Google Scholar] [CrossRef] [PubMed] - Bhattacharya, S.; John, S. Designing high-efficiency thin silicon solar cells using parabolic-pore photonic crystals. Phys. Rev. Appl.
**2018**, 9, 044009. [Google Scholar] [CrossRef] [Green Version] - Chang, W.H.; Chen, W.Y.; Chang, H.S.; Hsieh, T.P.; Chyi, J.I.; Hsu, T.M. Efficient Single-Photon Sources Based on Low-Density Quantum Dots in Photonic-Crystal Nanocavities. Phys. Rev. Lett.
**2006**, 96, 117401. [Google Scholar] [CrossRef] [PubMed] - Dosovitskiy, A.; Beyer, L.; Kolesnikov, A.; Weissenborn, D.; Zhai, X.; Unterthiner, T.; Dehghani, M.; Minderer, M.; Heigold, G.; Gelly, S.; et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv
**2020**, arXiv:2010.11929. [Google Scholar] - Vaswani, A.; Shazeer, N.; Parmar, N.; Uszkoreit, J.; Jones, L.; Gomez, A.N.; Kaiser, Ł.; Polosukhin, I. Attention is all you need. Adv. Neural Inf. Process. Syst.
**2017**, 30. [Google Scholar] - Hessel, M.; Modayil, J.; Van Hasselt, H.; Schaul, T.; Ostrovski, G.; Dabney, W.; Horgan, D.; Piot, B.; Azar, M.; Silver, D. Rainbow: Combining improvements in deep reinforcement learning. In Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, New Orleans, LA, USA, 2–7 February 2018. [Google Scholar]

**Figure 1.**The DNN model (LRS-RCNN) to correlate the input (i.e., design parameters) with output (i.e., optical parameters such as Q and V) of nanophotonic structures. LRS-RCNN consists of two convolutional layers (Conv1 and Conv2), each accompanied by an averaging pooling (AP) operation, and lastly three FC layers. LRS-RCNN tackles the prediction of optical properties as a deep learning regression problem. The detailed specifications of the network and its hyperparameters are marked on the diagram and/or listed in Table 1.

**Figure 2.**Pictorial illustration of the working principle of convolutional neural networks (CNN) with average pooling. Moreover, the formula for calculating the output feature size of a convolutional operation is also shown.

**Figure 3.**LRS-RCNN algorithm for modeling nanophotonic structures, implemented with a multi-step process: choose the desired structure, randomly fluctuate its design parameters and compute the optical properties in FDTD (pre-training steps), preprocess the dataset (step 1), train the model (step 2), and finally validate the model (step 3). Not shown in the diagram are the adaptive learning rates and L2 regularization, which are both core components of this algorithm.

**Figure 4.**Nanobeam structure. (

**a**) SEM image of an actual fabricated nanobeam by us. (

**b**,

**c**) The initial structure’s design parameters and optical properties, respectively. As shown, holes are elliptically shaped with ${r}_{1}$ and ${r}_{2}$ as its two axes. (

**d**) A sample in the dataset with randomly shifted holes. Blue: original holes; red: shifted holes.

**Figure 5.**L3 nanocavity structure. (

**a**) The initial structure’s design parameters and optical properties. Yellow box is the region where we randomly fluctuate the design parameters. Inset: the ${E}_{y}$ field of the initial structure. (

**b**,

**c**) Two samples in the dataset with randomly shifted holes. Black: original holes; red: shifted holes. Far right figures are magnified views of the configuration.

**Figure 6.**Learning curves of LRS-RCNN when trained to characterize nanobeam’s Q and V. Top panel contains results for Q: (

**a**) prediction error ${\u03f5}_{pred}$ vs. Epochs curve; (

**b**) $MSE$ vs. Epochs curve, for both training (yellow) and test (green) datasets. Min ${\u03f5}_{pred}$ and $MSE$ using the test dataset are also labeled on the figures, respectively. Bottom panel are the same results for V (

**c**,

**d**).

**Figure 7.**Validation results of the trained LRS-RCNN for nanobeam using the validation dataset. Plotted is the histogram of ${\u03f5}_{pred}$ for Q in (

**a**) and V in (

**b**). The boxplot in (

**c**) is extracted from data in (

**a**), and (

**d**) from (

**b**). Inset data: minimum, average, and median ${\u03f5}_{pred}$ values in the histograms.

**Figure 8.**Learning curves of LRS-RCNN when trained to characterize L3 nanocavity’s Q and V. Top panel contains results for Q: (

**a**) prediction error ${\u03f5}_{pred}$ vs. Epochs curve; (

**b**) $MSE$ vs. Epochs curve, for both training (yellow) and test (green) datasets. Min ${\u03f5}_{pred}$ and $MSE$ using the test dataset are also labeled on the figures, respectively. Bottom panel are the same results for V (

**c**,

**d**).

**Figure 9.**Validation results of the trained LRS-RCNN for L3 nanocavity using the validation dataset. Plotted is the histogram of ${\u03f5}_{pred}$ for Q in (

**a**) and V in (

**b**). The boxplot in (

**c**) is extracted from data in (

**a**), and (

**d**) from (

**b**). Inset data: minimum, average, and median ${\u03f5}_{pred}$ values in the histograms.

**Figure 10.**Results of training LRS-RCNN for L3 nanocavity when no LRS is used. The figure has the same order as Figure 8. One can observe the substantially noisier convergence curves in (

**a**–

**d**) in this case compared to before.

**Figure 11.**Results of training LRS-RCNN for L3 when no L2 regularization is used. The figure has the same order as Figure 8. It is observed that large overfittings are present in (

**a**–

**d**) in this case.

Hyperparameter | Value | Hyperparameter | Value |
---|---|---|---|

Conv1 | 20 3 × 3 kernels + 1 × 2 AP | FC1 | 240 neurons |

Conv2 | 40 3 × 3 kernels + 1 × 2 AP | FC2 | 120 neurons |

Paddings | 1 | FC3 | 50 neurons |

Activation function | ReLU | No. of Epochs | 700 |

Training batch size | 64 | Test batch size | 100 |

Optimizer | SGD | Initial learning rate | 0.01 |

Momentum | 0.5 | L2 regularization $\lambda $ | 0.001 |

Nanobeam’s $(H$, $W)$ | (1, 13) | Learning rate scheduler | ReduceOnPlateau |

L3’s $(H$, $W)$ | (5, 12) | Loss function | $MSE$ |

**Table 2.**Specifications of the dataset, generated according to Section 2.2, for training LRS-RCNN (DA = design parameter, OP = optical property, and DOF = degree of freedom). Left panel: nanobeam; right panel: L3 nanocavity. DAs such as “x locations” and “r” refer to the x coordinate and radius of each individual air hole, respectively. Corresponding OPs are computed by FDTD simulation.

Nanobeam DAs | Nanobeam OPs | L3 DAs | L3 OPs |
---|---|---|---|

${r}_{1}$ | Q-factor | x locations | Q-factor |

${r}_{2}$ | Modal volume V | y locations | Modal volume V |

x locations | r | ||

$DOF=13\times 3=39$ | $DOF=54\times 3=162$ |

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## Share and Cite

**MDPI and ACS Style**

Li, R.; Gu, X.; Shen, Y.; Li, K.; Li, Z.; Zhang, Z.
Smart and Rapid Design of Nanophotonic Structures by an Adaptive and Regularized Deep Neural Network. *Nanomaterials* **2022**, *12*, 1372.
https://doi.org/10.3390/nano12081372

**AMA Style**

Li R, Gu X, Shen Y, Li K, Li Z, Zhang Z.
Smart and Rapid Design of Nanophotonic Structures by an Adaptive and Regularized Deep Neural Network. *Nanomaterials*. 2022; 12(8):1372.
https://doi.org/10.3390/nano12081372

**Chicago/Turabian Style**

Li, Renjie, Xiaozhe Gu, Yuanwen Shen, Ke Li, Zhen Li, and Zhaoyu Zhang.
2022. "Smart and Rapid Design of Nanophotonic Structures by an Adaptive and Regularized Deep Neural Network" *Nanomaterials* 12, no. 8: 1372.
https://doi.org/10.3390/nano12081372