超穎表面具有特殊的次波長微結構，使其能在極短的光學距離內精準調控電磁波之相位與振幅，而全介電質超穎表面單元便是其中一種極具潛力之設計。當光波與這些高折射率奈米柱相互作用時，會激發強烈的米氏共振，進而產生高度非線性之光學響應。在現今，超穎表面已廣泛應用於許多前瞻領域。例如，在消費性電子中，它能用來設計輕薄的超穎透鏡；在自動駕駛領域，它有助於全固態光達的光束轉向技術發展。此外，其微觀幾何圖形引發之繞射現象，亦與半導體先進製程中的運算微影面臨高度相似的物理挑戰。
本研究建構了一套結合物理資訊神經網路（Physics-Informed Neural Network, PINN）與外部數值分析數據之混合模擬架構，模擬並分析電磁波分別在基礎波導結構與全介電質超穎表面陣列中之傳播行為。
本研究在波導結構中引入有限元素分析（Finite Element Method, FEM）之數據，用以驗證模型之準確度；而在超穎表面陣列中，引入嚴格耦合波分析（Rigorous Coupled-Wave Analysis, RCWA）之數據作為數據錨點，成功預測共振振幅並且建構出二維光學相位與穿透率之資料庫。

Metasurfaces feature unique sub-wavelength microstructures that enable precise control over the phase and amplitude of electromagnetic waves within an extremely short optical distance. Among them, the all-dielectric metasurface unit represents a highly promising design. When light interacts with these high-refractive-index nano-pillars, strong Mie resonances are excited, resulting in highly nonlinear optical responses. Currently, metasurfaces are widely applied in numerous cutting-edge fields. For instance, in consumer electronics, they are utilized to design ultra-thin metalenses; in autonomous driving, they facilitate the development of beam steering technologies for solid-state LiDAR. Furthermore, the diffraction phenomena induced by their microscopic geometries face physical challenges highly analogous to computational lithography in advanced semiconductor manufacturing.
This study constructs a hybrid simulation framework integrating Physics-Informed Neural Networks (PINNs) with external numerical analysis data to simulate and analyze the propagation behaviors of electromagnetic waves in a basic waveguide structure and an all-dielectric metasurface array, respectively.
In the waveguide structure, Finite Element Method (FEM) data is introduced to verify the accuracy of the model. In the metasurface array, Rigorous Coupled-Wave Analysis (RCWA) data is introduced as data anchors, which successfully predicts resonance amplitudes and constructs a high-precision two-dimensional optical phase and transmission library.

[1] Yu, N., Genevet, P., Kats, M. A., Aieta, F., Tetienne, J. P., Capasso, F., & Gaburro, Z. (2011). Light propagation with phase discontinuities: generalized laws of reflection and refraction. Science, 334(6054), 333-337.
[2] Raissi, M., Perdikaris, P., & Karniadakis, G. E. (2019). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686-707.
[3] Peurifoy, J., Choi, Y., Huguenin-Dumittan, C., Fei, N. C., Amsellem, J., ... & Soljačić, M. (2018). Nanophotonic particle simulation and inverse design using artificial neural networks. Science Advances, 4(6), eaar4206.
[4] Jackson, J. D. (1998). Classical Electrodynamics (3rd ed.). Wiley.
[5] Born, M., & Wolf, E. (1999). Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (7th ed.). Cambridge University Press.
[6] Zienkiewicz, O. C., Taylor, R. L., & Zhu, J. Z. (2005). The Finite Element Method: Its Basis and Fundamentals (6th ed.). Elsevier.
[7] Béchet, E., Cuillière, J. C., & Trochu, F. (2002). Generation of a finite element MESH from stereolithography (STL) files. Computer-Aided Design, 34(1), 1-17.
[8] Moharam, M. G., & Gaylord, T. K. (1981). Rigorous coupled-wave analysis of planar-grating diffraction. Journal of the Optical Society of America, 71(7), 811-818.
[9] Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021). Physics-informed machine learning. Nature Reviews Physics, 3(6), 422-440.
[10] Chen, Y., Lu, L., Karniadakis, G. E., & Dal Negro, L. (2020). Physics-informed neural networks for inverse problems in nano-optics and metamaterials. Optics Express, 28(8), 11618-11633.
[11] Lim, J., & Psaltis, D. (2022). MaxwellNet: Physics-driven deep neural network training based on Maxwell’s equations. APL Photonics, 7(1), 011301.
[12] Lim, J., & Psaltis, D. (2023). Predicting nonlinear optical scattering with physics-driven neural networks. APL Photonics, 8(2), 026105.
[13] Zhang, Y., et al. (2022). Physics-Informed Deep Neural Network for Inhomogeneous Magnetized Plasma Parameter Inversion. IEEE Antennas and Wireless Propagation Letters, 21(4), 798-802.
[14] Jarolim, R., et al. (2023). Probing the solar coronal magnetic field with physics-informed neural networks. Nature Astronomy, 7(10), 1171-1179.
[15] Lu, L., Meng, X., Mao, Z., & Karniadakis, G. E. (2021). DeepXDE: A deep learning library for solving differential equations. SIAM Review, 63(1), 208-228.
[16] Kuznetsov, A. I., Miroshnichenko, A. E., Brongersma, M. L., Kivshar, Y. S., & Luk'yanchuk, B. (2016). Optically resonant dielectric nanostructures. Science, 354(6314), aag2472.
[17] Wang, S., Teng, Y., & Perdikaris, P. (2021). Understanding and mitigating the spectral bias of physics-informed neural networks. ICML 2021.
[18] Sitzmann, V., Martel, J. N., Bergman, A. W., Lindell, D. B., & Wetzstein, G. (2020). Implicit neural representations with periodic activation functions. Advances in Neural Information Processing Systems (NeurIPS), 33, 7493-7504.
[19] The MathWorks, Inc. (2024). MATLAB Documentation. Natick, MA: The MathWorks, Inc.
[20] Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep Learning. MIT Press.
[21] Baydin, A. G., Pearlmutter, B. A., Radul, A. A., & Siskind, J. M. (2018). Automatic differentiation in machine learning: a survey. Journal of Machine Learning Research, 18(153), 1-43.
[22] Willmott, C. J., & Matsuura, K. (2005). Advantages of the Mean Absolute Error (MAE) over the Root Mean Square Error (RMSE) in assessing average model performance. Climate Research, 30(1), 79-82.
[23] Kingma, D. P., & Ba, J. (2015). Adam: A Method for Stochastic Optimization. International Conference on Learning Representations (ICLR).
[24] DeepXDE Documentation, Version 1.13.2. Retrieved July 15, 2026, from https://deepxde.readthedocs.io/en/v1.13.2/