A Regularized Procedure to Generate a Deep Learning Model for Topology Optimization of Electromagnetic Devices

The use of behavioral models based on deep learning (DL) to accelerate electromagnetic field computations has recently been proposed to solve complex electromagnetic problems. Such problems usually require time-consuming numerical analysis, while DL allows achieving the topologically optimized design of electromagnetic devices using desktop class computers and reasonable computation times. An unparametrized bitmap representation of the geometries to be optimized, which is a highly desirable feature needed to discover completely new solutions, is perfectly managed by DL models. On the other hand, optimization algorithms do not easily cope with high dimensional input data, particularly because it is difficult to enforce the searched solutions as feasible and make them belong to expected manifolds. In this work, we propose the use of a variational autoencoder as a data regularization/augmentation tool in the context of topology optimization. The optimization was carried out using a gradient descent algorithm, and the DL neural network was used as a surrogate model to accelerate the resolution of single trial cases in the due course of optimization. The variational autoencoder and the surrogate model were simultaneously trained in a multi-model custom training loop that minimizes total loss—which is the combination of the two models’ losses. In this paper, using the TEAM 25 problem (a benchmark problem for the assessment of electromagnetic numerical field analysis) as a test bench, we will provide a comparison between the computational times and design quality for a “classical” approach and the DL-based approach. Preliminary results show that the variational autoencoder manages regularizing the resolution process and transforms a constrained optimization into an unconstrained one, improving both the quality of the final solution and the performance of the resolution process.

[1]  Hajime Igarashi,et al.  Transfer Learning Through Deep Learning: Application to Topology Optimization of Electric Motor , 2020, IEEE Transactions on Magnetics.

[2]  Arbaaz Khan,et al.  Deep Learning for Magnetic Field Estimation , 2019, IEEE Transactions on Magnetics.

[3]  N. Takahashi,et al.  Investigation of simulated annealing method and its application to optimal design of die mold for orientation of magnetic powder , 1996 .

[4]  Max Welling,et al.  Auto-Encoding Variational Bayes , 2013, ICLR.

[5]  Francesco Cupertino,et al.  A Design Method for the Cogging Torque Minimization of Permanent Magnet Machines with a Segmented Stator Core Based on ANN Surrogate Models , 2021, Energies.

[6]  Sebastian Schöps,et al.  Deep Learning-Based Prediction of Key Performance Indicators for Electrical Machines , 2021, IEEE Access.

[7]  Jos'e Miguel Hern'andez-Lobato,et al.  Sample-Efficient Optimization in the Latent Space of Deep Generative Models via Weighted Retraining , 2020, NeurIPS.

[8]  Aaron C. Courville,et al.  Generative adversarial networks , 2020 .

[9]  Jaime Lloret,et al.  Conditional Variational Autoencoder for Prediction and Feature Recovery Applied to Intrusion Detection in IoT , 2017, Sensors.

[10]  Ikjin Lee,et al.  Deep Generative Design: Integration of Topology Optimization and Generative Models , 2019, Journal of Mechanical Design.

[11]  Chris Yakopcic,et al.  A State-of-the-Art Survey on Deep Learning Theory and Architectures , 2019, Electronics.

[12]  Nurshazlyn Mohd Aszemi,et al.  Hyperparameter Optimization in Convolutional Neural Network using Genetic Algorithms , 2019, International Journal of Advanced Computer Science and Applications.

[13]  Sara Carcangiu,et al.  Grid-Enabled Tabu Search for Electromagnetic Optimization Problems , 2010, IEEE Transactions on Magnetics.

[14]  Towards End-to-End Deep Learning Performance Analysis of Electric Motors , 2021 .

[15]  Hyung-Jeong Yang,et al.  Survival Prediction of Lung Cancer Using Small-Size Clinical Data with a Multiple Task Variational Autoencoder , 2021, Electronics.

[16]  Hajime Igarashi,et al.  Topology Optimization Accelerated by Deep Learning , 2019, IEEE Transactions on Magnetics.

[17]  Jimmy Ba,et al.  Adam: A Method for Stochastic Optimization , 2014, ICLR.

[18]  Jianguo Zhu,et al.  Machine Learning for Design Optimization of Electromagnetic Devices: Recent Developments and Future Directions , 2021, Applied Sciences.

[19]  Barak A. Pearlmutter,et al.  Automatic differentiation in machine learning: a survey , 2015, J. Mach. Learn. Res..

[20]  Mauro Tucci,et al.  Autoencoder Based Optimization for Electromagnetics Problems , 2019 .

[21]  Frank Noé,et al.  Efficient multi-objective molecular optimization in a continuous latent space† †Electronic supplementary information (ESI) available: Details of the desirability scaling functions, high resolution figures and detailed results of the GuacaMol benchmark. See DOI: 10.1039/c9sc01928f , 2019, Chemical science.

[22]  Alessandro Formisano,et al.  A Deep Learning Surrogate Model for Topology Optimization , 2021, IEEE Transactions on Magnetics.

[23]  Yi Li,et al.  A Robust-Equitable Measure for Feature Ranking and Selection , 2017, J. Mach. Learn. Res..

[24]  Arbaaz Khan,et al.  Efficiency Map Prediction of Motor Drives Using Deep Learning , 2020, IEEE Transactions on Magnetics.

[25]  Kangfeng Zheng,et al.  Improving the Classification Effectiveness of Intrusion Detection by Using Improved Conditional Variational AutoEncoder and Deep Neural Network , 2019, Sensors.

[26]  Diederik P. Kingma,et al.  An Introduction to Variational Autoencoders , 2019, Found. Trends Mach. Learn..

[27]  Luca Sani,et al.  Deep Learning and Reduced Models for Fast Optimization in Electromagnetics , 2020, IEEE Transactions on Magnetics.

[28]  H. Igarashi,et al.  Multi-Objective Topology Optimization of Rotating Machines Using Deep Learning , 2019, IEEE Transactions on Magnetics.

[29]  Roi Livni,et al.  On the Computational Efficiency of Training Neural Networks , 2014, NIPS.

[30]  Jiefu Chen,et al.  A Maxwell's Equations Based Deep Learning Method for Time Domain Electromagnetic Simulations , 2020, 2020 IEEE Texas Symposium on Wireless and Microwave Circuits and Systems (WMCS).