Machine learning for knowledge acquisition and accelerated inverse-design for non-Hermitian systems

Machine learning for knowledge acquisition and accelerated inverse-design for non-Hermitian systems

Item Type Article

Authors Ahmed, Waqas Waseem;Farhat, Mohamed;Staliunas, Kestutis;Zhang, Xiangliang;Wu, Ying

Citation Ahmed, W. W., Farhat, M., Staliunas, K., Zhang, X., & Wu, Y. (2023).

Machine learning for knowledge acquisition and accelerated inverse-design for non-Hermitian systems. Communications Physics, 6(1). https://doi.org/10.1038/s42005-022-01121-9 Eprint version Publisher's Version/PDF

DOI 10.1038/s42005-022-01121-9

Publisher Springer Science and Business Media LLC Journal Communications Physics

Rights Archived with thanks to Communications Physics under a Creative Commons license, details at: https://creativecommons.org/

licenses/by/4.0 Download date 2023-10-31 02:39:50

Item License https://creativecommons.org/licenses/by/4.0

Link to Item http://hdl.handle.net/10754/676744

Supplementary Information

Machine learning for knowledge acquisition and accelerated inverse-design for non-Hermitian systems

Waqas W. Ahmed

¹

, Mohamed Farhat

¹

, Kestutis Staliunas

^2,3,4

, Xiangliang Zhang

^1,5†

, and Ying Wu

^1,6*

1

Division of Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

2

Departament de Física, Universitat Politècnica de Catalunya (UPC), Colom 11, E-08222 Terrassa, Barcelona, Spain

3

Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig Lluís Companys 23, E- 08010, Barcelona, Spain

4

Vilnius University, Laser Research Center, Saulėtekio al. 10, Vilnius, Lithuania

5

Department of Computer Science and Engineering, University of Notre Dame, Notre Dame, IN 46556, United States of America

6

Division of Physical Science and Engineering, King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Saudi Arabia

Emails:

^*

ying.wu@kaust.edu.sa,

^†

xzhang33@nd.edu

Supplementary Note 1: Transmission prediction from left reflection

In the main article, the neural network is proposed to provide one-to-one mapping from left reflection to transmission spectra. More examples of our designed network to generate the transmission profile from left reflection are provided in Supplementary Figure 1.

Supplementary Figure 1: Results of the feed forward neural network for mapping from left reflections to transmission response. a Decrease of loss function curve when training the neural network. b Histogram of spectral prediction error. c Representative examples for generation of transmission response from given right reflection spectrum (blue curve). The solid red and dotted black line represent the transmission calculated from TMM and ML method, respectively. The neural architecture is the same as the one in the main text. Rectified linear unit activation function is used for each neural. The Adam optimizer is used for training the neural network, with a learning rate 5×10^-4.

Supplementary Note 2: Transmission prediction from right reflection

We show how the transmission spectrum is generated from the left reflection in the main text.

Right reflection and transmission, on the other hand, have also a one-to-one correspondence. The architecture is identical, except during training, right reflection is used as input. The network's training and prediction performance are presented in Supplementary Figure 2a, b. Supplementary Figure 2c shows some examples of predicted (dotted black) and target (solid black) responses (red solid).

Supplementary Figure 2: Training and prediction performance of the designed neural network for right reflection to transmission mapping. a Decrease of loss function curve when training the neural network. b Histogram of spectral prediction error. c Representative examples for generation of transmission response from given right

reflection spectrum (blue curve). The solid red and dotted black line represent the transmission calculated from TMM and ML method, respectively. The neural architecture is the same as the one in the main text.

Supplementary Note 3: Transmission spectrum prediction from given right and left reflection

We may alternatively generate the transmission profile by feeding both left and right reflection information to the network in a more basic manner. The number of input units are then doubled in terms of network architecture, while the rest of the neural network layers remains the same. Due to the extra information in the input data, the relative prediction error is reduced more than for either left or right reflection input. The results are summarized in Supplementary Figure 3.

Supplementary Figure 3: Results of the designed network for mapping both left and right reflection to transmission response. a Decrease of loss function curve when training the neural network. b Histogram of spectral

prediction error. c Representative examples for generation of transmission response from given right (blue curve) and left (magenta curve) reflection spectra. The solid red and dotted black line represent the transmission calculated from TMM and ML method, respectively.

Supplementary Note 4: Examples of Forward Network

More examples of the designed forward model covered in the main article are presented in Supplementary Figure 4. The design parameters of the predicted responses are provided in Supplementary Table 1.

Supplementary Figure 4: Results of designed forward network to predict the spectral responses from the given design parameters. a-c Representative examples of the predicted spectral response for the designed networks (i) transmission, (ii) left reflection, and (iii) right reflection.

Supplementary Table 1: Corresponding design parameters for the predicted responses using forward model.

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Fig.4c

^{1.34 1.29 1.2 1.09 1.32 -0.13 0.11 0.045} -0.015 -0.075

Fig.4d

^{1.12 1.26 1.18 1.3 1.06} -0.115 -0.005 0.015 -0.105 0.165

Fig.S4a

^{1.01 1.31 1.29 1.36 1.21 -0.03 0 0.1 -0.19 0.08}

Fig.S4b

^{1.38 1.02 1.31 1.35 1.15 0.185 -0.2 -0.04 -0.18 -005}

Fig.S4c

^{1.06 1.16 1.07 1.33 1.38} -0.045 0.075 0.03 -0.195 0.115

Supplementary Note 5: Examples of Inverse Network

More examples of the designed inverse model covered in the main article are presented in

Supplementary Figure 5. The design parameters of the generated responses are provided in

Supplementary Table 2.

Supplementary Figure 5: Results of designed inverse network to predict the design parameters from the given spectral response. a-c Representative examples of the predicted spectral response for the designed networks (i) transmission, (ii) left reflection, and (iii) right reflection.

Supplementary Table 2: Corresponding design parameters for the generated responses using inverse model.

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^{1.26 1.27 1.14 1.16 1.025 -0.0938 -0.039 -0.065 -0.0428 -0.1025}

Fig.5c

^{1.1 1.15 1.2 1.05 1 -0.0749} -0.0587 -0.060 0.0964 -0.0231

Fig.5c

^{1.24 1.67 1.438 1.26 1.18 0.143 -0.151 0.174 0.0388 -0.0639}

Fig.S5a

^{1.196 1.35 1.38 1.33 1.203 -0.0277} -0.0744 -0.0808 -0.1178 0.0998

Fig.S5b

^{1.049 0.96 1.31 1.342 1.375 0.0271} -0.0732 -0.0846 0.06 0.2505

Fig.S5c

^{0.934 1.301 1.393 1.302 1.23 -0.069 0.0588 -0.0916 0.0269 -0.0266}