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A rigorous physics-informed learning methodology is proposed for predictions of wave solutions and band structures in electronic and optical superlattice structures. The methodology is enabled by proper orthogonal decomposition (POD) and Galerkin projection of the wave equation. The approach solves the wave eigenvalue problem in POD space constituted by a finite set of basis functions (or POD modes). The POD ensures that the generated modes are optimized and tailored to the parametric variations of the system. Galerkin projection however enforces physical principles in the methodology to further enhance the accuracy and efficiency of the developed model. It has been demonstrated that the POD-Galerkin methodology offers an approach with a reduction in degrees of freedom by 4 orders of magnitude, compared to direct numerical simulation (DNS). A computing speedup near 15,000 times over DNS can be achieved with high accuracy for either of the superlattice structures if only the band structure is calculated without the wave solution. If both wave function solution and band structure are needed, a 2-order reduction in computational time can be achieved with a relative least square error (LSE) near 1%. When the training is incomplete or the desired eigenstates are slightly beyond the training bounds, an accurate prediction with an LSE near 1%-2% still can be reached if more POD modes are included. This reveals its remarkable learning ability to reach correct solutions with the guidance of physical principles provided by Galerkin projection.
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Fast Simulations with High Accuracy for Periodic Quantum Nanostructures and Photonic Crystals Enabled by a Physics-Informed Projection Learning Methodology
Fast solution of wave propagation in periodic structures usually relies on simplified approaches, such as analytical methods, transmission line models, scattering matrix approaches, plane wave methods, etc. For complex multi-dimensional problems, computationally intensive direct numerical simulation (DNS) is always needed. This study demonstrates a fast and accurate simulation methodology enabled by a physics-based learning methodology, derived from proper orthogonal decomposition (POD) and Galerkin projection, for periodic quantum nanostructure and photonic crystals. POD is a projection-based method that generates optimal basis functions (or POD modes) via solution data collected from DNSs. This process trains the POD modes to adapt parametric variations of the system and offers the best least squares (LS) fit to the solution using the smallest number of modes. This is very different from other projection approaches, e.g., Fourier, Legendre, Bessel, Airy functions, etc., that adopt assumed basis functions selected for the problem based on the solution form. After generating the optimal POD modes, Galerkin projection of the wave equation onto each of the POD modes is performed to close the model and incorporate physical principles guided by the wave equation. Such a rigorous approach offers efficient simulations with high accuracy and exhibits the extrapolation ability in cases reasonably beyond the training bounds. The POD-Galerkin methodology is applied in this study to predict band structures and wave solutions for 2D periodic quantum-dot and photonic-lattice structures. The plane-wave approach is also included in a periodic quantum-dot structure to illustrate the superior performance of the POD-Galerkin methodology. The POD-Galerkin approach offers a 2-order computing speedup for both nanostructure and optical superlattices, compared to DNS, when solving both the wave solution and band structure. If the band structure is the only concern, a 4-order improvement in computational efficiency can be achieved. Fig. 1(a) shows the optical superlattice in a demonstration, where a unit cell includes 22 discs with diagonally symmetrical refractive indices and the background index n = 1. The POD modes for this case are trained by TE mode electric field data collected from DNSs with variation of diagonally symmetrical refractive indices. The LS error of the predicted electric field wave solution from the POD-Galerkin approach, shown in Fig. 1(b) compared to DNS, is below 1% with just 8 POD modes that offer a more than 4-order reduction in the degrees of freedom, compared to DNS. In addition, an extremely accurate prediction of band structure is illustrated in Fig. 1(c) with a maximum error below 0.1% in the entire Brillouin zone.
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- Award ID(s):
- 2118079
- PAR ID:
- 10555799
- Publisher / Repository:
- XXXV IUPAP. Comp. Physics (CCP2024)
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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