Simulation methods for electromagnetic and optical structures

Abstract

The present invention relates to physics-informed projection learning methods for simulation of single-block and multi-block electromagnetic and optical structures. The methods of the invention offer fast and accurate multi-dimensional simulations for the time-dependent electromagnetic wave equation and the frequency-domain electromagnetic wave (Helmholtz) equation. Both equations can be expressed in terms of electric field, magnetic field or vector potential. The methods of the invention can be implemented in both frequency and time domains to predict electromagnetic field solutions for effective design and analysis of electromagnetic and photonic structures, including calculations of electromagnetic / optical band structures for periodic structures with or without imperfections or defects. These electromagnetic / photonic structures include, but not limited to, antennas, waveguides, interconnects, superlattices, metamaterials, photonic crystals, electromagnetic-bandgap structures, etc. Depending on the wavelength, they could be built in structures from nm to cm scales, such as nanostructured materials, semiconductor chips, hybrid integrated circuits (ICs), printed circuit boards, etc.

Description

SIMULATION METHODS FOR ELECTROMAGNETIC AND OPTICAL STRUCTURESCROSS-REFERENCE TO RELATED APPLICATION

[0001] The present application relates and claims priority to United States Provisional Application Serial Number 63 / 664,851, filed June 27, 2024, the entirety of which is incorporated by reference.BACKGROUND OF THE INVENTION1. FIELD OF INVENTION

[0002] The present disclosure is directed generally to simulation methods for electromagnetic and optical structures.2. BRIEF SUMMARY OF INVENTION

[0003] Demands for higher frequency electronic operations have drastically increased in the last century for all electronic applications, including communications, computing, control, etc., in nearly all industries. In general, higher operating frequency offers larger bandwidth, faster communication, increased efficiency, better signal and image quality, etc. The maximum allowed frequency on today ’ s semiconductor chips is however primarily limited by electronic interconnects that suffer from their physical limitations induced by parasitic resistance, capacitance and inductance. These parasitic elements lead to slower speeds, lower frequency, interference noise due to cross talk, higher energy consumption, higher chip temperature, etc. The desire to enable higher frequency operations in communications and computing systems has been pushing the realization of optical, photonic and optoelectronic technologies in nanostructures and semiconductor chips, including photonic devices, integrated circuits (ICs), interconnects and waveguides, using light to transmit / process signals and data. In addition to faster transmission speeds, higher frequency operations and larger bandwidths, implementation of photonic technologies on semiconductor chips could also remove or significantly minimize all the above- mentioned drawbacks of the conventional electronic interconnects. Moreover, recent advances in electromagnetic bandgap structures, photonic crystals, photonic superlattices, and electromagnetic / photonic metamaterials have provided new concepts to manipulate wave propagation and interactions in periodic electromagnetic / photonic structures with or without defects / imperfections. These could significantly enhance performance of electromagnetic / photonic devices, structures, antennas, and interconnects and widen theirapplications not only for photonic interconnects / ICs but also for lower frequency applications, such as terahertz wave, microwave and millimeter wave antennas, sensors and waveguides that may be built in semiconductor chips, hybrid ICs, printed circuit boards, etc.

[0004] Accurate design and analysis of electromagnetic or optical structures in general require multi-dimensional numerical solutions of electromagnetic fields. The solutions are determined by the electromagnetic wave equation that can be derived from the coupled equations of Maxwell Faraday’s and Ampere’s Laws. Any non-TEM (transverse electromagnetic) wave can be decomposed into transverse electric (TE) and transverse magnetic (TM) modes. The electromagnetic wave equation is in general expressed in terms of electric field for TE waves and magnetic field for TM waves. To simplify the expression, the electromagnetic wave equation can also be written in terms of (magnetic) vector potential with a selected gauge, such as the Lorenz or Coulomb gauge. Although use of vector potential may simplify the electromagnetic wave equation in some applications, it needs additional calculations when electric or magnetic field solution is needed. In particular, if the electric field is needed from the (magnetic) vector potential, it is necessary to solve an additional wave equation for the (electric) scalar potential when using Lorenz gauge. When using the Coulomb gauge, a wave equation in terms of both vector and scalar potentials coupled with Poisson’s equation for potential needs to be solved together, which is not included in this invention.

[0005] For simple structures, electromagnetic wave solutions may be approximated by efficient compact models, such as analytical, transmission-line, scattering-matrix or transfermatrix methods. If accurate solutions for electromagnetic fields in complex multi-dimensional structures are needed, computationally intensive direct numerical simulations (DNSs) with fine spatial resolution are always needed. Most popular and accurate commercial and open-source electromagnetic simulation tools are based on DNS methods, such as finite-difference, finite element or finite volume methods. This invention provides physics-informed learning simulation methods that offer efficient spatial electromagnetic field solutions in time and frequency domains with high accuracy and fine resolution. The invention also offers extremely fast and accurate calculations of electromagnetic / optical band structures for periodic electromagnetic / optical structures, such as superlattices, photonic crystals and metamaterials with or without defects or imperfections. The electric / magnetic field and band structure solutions obtained from the methods of the invention can be further applied for efficient and accurate calculations / analysis of propertiesneeded for design of electromagnetic / photonic devices and structures, for example, the characteristics impedance, dispersion relations, reflection / transmission coefficient, S-parameters, etc. The invention provides cost-effective tools for design of various electromagnetic, optical and photonic devices / structures and for analysis of their device characteristics and performance.

[0006] According to an aspect is a computer-implemented block-based projection learning method for time-domain simulation of optical or electromagnetic devices, structure or system derived from physics-informed learning algorithms comprising the steps of executing, by a processor of a computing system, direct numerical simulations of the dynamic electromagnetic wave equation for each selected block of the device, structure, or system; temporarily storing, in random access memory, numerical data generated during the direct numerical simulation for each selected block; storing, in non-volatile memory, training data derived from the direct numerical simulation for each selected block; generating, by the processor, basis functions and eigenvalues from the training data of each block to represent a reduced-dimensional space for the block, using a learning algorithm; performing Galerkin projection of the time-dependent electromagnetic wave equation for each selected block onto the reduced-dimensional space to construct a set of ordinary differential equations for the block in the reduced-dimensional space; enforcing the continuity for subdomain blocks at the interfaces between the selected block and variation of adjoining blocks by an interface smoothing technique; calculating, by the processor, model parameters for the ODEs for each selected block in the reduced-dimensional space including interface coupling coefficients influenced by variations of adjoining blocks resulting from the smoothing technique; and storing, in a database, the basis functions, eigenvalues and model parameters for each selected block.

[0007] According to an embodiment, the method further comprises constructing a singleblock dynamic projection-based learning model using a selected block stored in the database for a single-block structure.

[0008] According to an embodiment, the method further comprises constructing a multiblock dynamic projection-based learning model using selected blocks stored in the database for a multi-block structure with interface continuities enforced by a smoothing technique.

[0009] According to an embodiment, the electromagnetic wave equation can be expressed in terms of either electric, magnetic field, magnetic vector potential, or electric scalar potential, and optionally applying the Lorenz gauge during training data collection and Galerkin projection.

[0010] According to an embodiment, the projection learning algorithm is derived from one or more of: reduced order learning methods, data grouping / clustering techniques, neural networks, machine learning or artificial intelligence.

[0011] According to an embodiment, the block-based learning can be adapted to simulate devices, structures, systems and blocks with isotropic, anisotropic, uniform, nonuniform, linear and nonlinear materials.

[0012] According to an embodiment, the method further comprises performing simulation in the reduced-dimensional space for a device, structure or system, that consists of one or more blocks with one or more basis functions for each block using the interface coefficients to ensure continuity at each interface between adjoining blocks with appropriate boundary coefficients for the entire structure.

[0013] According to an embodiment, the method further comprises performing post processing to calculate the spatiotemporal solutions for one or more electric field, magnetic field, scalar potential or vector potential, wherein the electric field is calculated from scalar and vector potentials when the vector potential and scalar potential are solved from their wave equations in their reduced dimensional spaces using Lorenz gauge.

[0014] According to an embodiment, the method further comprises using the postprocessed results to calculate properties of the optical or electromagnetic devices, structures, or systems by coupling the projection-based learning models with other models or simulators to perform multi-physics simulations to further calculate properties of the said devices, structures, or systems, wherein the properties include at least one of the following: characteristic impedance, dispersion relations, reflection and transmission coefficient, and S-parameters.

[0015] According to an embodiment, the magnetic field is calculated using pre-calculated curl data of electric-field modes based on Maxwell-Faraday’s law, and the electric field is calculated using precalculated data of magnetic field modes based on Maxwell-Ampere’s law.

[0016] According to an embodiment, the method further comprises pre-calculating the curl of vector-potential modes and the gradient of scalar potential when solving the vector-potential and scalar-potential wave equations in their reduced dimensional spaces.

[0017] According to an embodiment, the method further comprises applying the said projection-based learning algorithm to consolidate smaller-size block models to generate a projection-based learning model for a larger-size block.

[0018] According to another aspect the invention is a computer-implemented block-based projection learning method for frequency-domain simulation of optical or electromagnetic devices, structure or system derived from physics-informed learning algorithms based on Bloch’s theorem comprising the steps of: executing, by a processor of a computing system, direct numerical simulations of the dynamic electromagnetic wave equation in the frequency domain for each selected block at each selected k point in the Brillouin zone of the device, structure, or system; temporarily storing, in random access memory, numerical data generated during the direct numerical simulation for each selected block; storing, in non-volatile memory, training data derived from the direct numerical simulation for each selected block; generating, by the processor, basis functions and eigenvalues for each selected k point from the training data of each block to represent the reduced-dimensional space for the block, using a learning algorithm; performing Galerkin projection of the electromagnetic wave equation for each selected block in the frequency domain onto the reduced-dimensional space to construct an eigenvalue matrix equation in the reduced-dimensional space at each k point for the block; enforcing the continuity for subdomain blocks at the interfaces between the selected block and variation of adjoining blocks by an interface smoothing technique; calculating, by the processor, model parameters for the eigenvalue matrix equation for each selected block at each selected kpoint; and storing, in a database, the basis functions, eigenvalues and all model parameters at each k point for each selected block.

[0019] According to an embodiment, a structure or subdomain is subjected to parametric variations at each selected k point in the Brillouin zone of the reciprocal lattice space for said structure or subdomain.

[0020] According to an embodiment, the model parameters include interface coupling coefficients that are derived using an interface smoothing technique and account for variations in adjoining blocks.

[0021] According to an embodiment, the method further comprises constructing a singleblock dynamic projection-based learning model in the frequency domain using a selected block stored in the database for a single-block structure.

[0022] According to an embodiment, the method further comprises constructing a multiblock dynamic projection-based learning model in the frequency domain using selected blocks stored in the database for a multi-block structure with interface continuities enforced by a smoothing technique.

[0023] According to an embodiment, the method further comprises the electromagnetic wave equation can be expressed in terms of either electric, magnetic field, magnetic vector potential, or electric scalar potential, and optionally applying the Lorenz gauge during training data collection and Galerkin projection.

[0024] According to an embodiment, the projection learning algorithm is derived from one or more of: reduced order learning methods, data-grouping / clustering techniques, neural networks, machine learning or artificial intelligence.

[0025] According to an embodiment, the block-based learning can be adapted to simulate devices, structures, systems and blocks with isotropic, anisotropic, uniform, nonuniform, linear and nonlinear materials.

[0026] According to an embodiment the method further comprises the generation of band structure in the Brillouin zone from the eigenfrequencies at each k point.

[0027] According to an embodiment the method further comprises performing simulation of the eigenvalue matrix equation at each k point in the reduced-dimensional space to calculate eigenfrequencies and eigenvectors for a device, structure or system that consists of one or more blocks with one or more basis functions for each block using interface coefficients to ensure continuity at each interface between adjoining blocks with appropriate boundary coefficients for a structure.

[0028] According to an embodiment the method further comprises performing postprocessing using the eigenvectors to calculate on or more of: the spatial solution of electric field, magnetic field, scalar potential, or vector potential for each frequency state at selected k points, wherein the Lorenz gauge is used to calculate the spatial electric field from scalar and vector potentials at the selected k points.

[0029] According to an embodiment the method further comprises coupling the said projection-based learning models in the frequency domain with other models or tools to perform multi-physics simulations to further calculate properties of the said devices, structures, or systems, wherein the properties include at least one of the following: characteristic impedance, dispersion relations, reflection and transmission coefficient, and S-parameters.

[0030] According to an embodiment the method further comprises applying the said projection-based learning algorithm to consolidate smaller-size block models to generate a projection-based learning model for a larger-size block.

[0031] According to an embodiment the magnetic field is calculated using pre-calculated curl data of electric-field modes based on Maxwell-Faraday’s law, and the electric field is calculated using precalculated data of magnetic field modes based on Maxwell-Ampere’s law.

[0032] According to an embodiment the method further comprises the magnetic field is calculated using pre-calculated curl data of electric-field modes based on Maxwell-Faraday’s law, and the electric field is calculated using precalculated data of magnetic field modes based on Maxwell- Ampere’s law.

[0033] These and other aspects of the invention will be apparent from the embodiments described below.BRIEF DESCRIPTION OF THE DRAWINGS

[0034] The present invention will be more fully understood and appreciated by reading the following Detailed Description in conjunction with the accompanying drawings, in which:

[0035] FIG. 1 is a flow chart illustrating a method to construct a single-block physics- informed reduced-order learning model for simulations of electromagnetic / optical structures in the time domain, in accordance with an aspect of the invention.

[0036] FIG. 2 is a flow chart illustrating a method to construct a multi-block physics- informed reduced-order learning model for simulations of electromagnetic / optical structures in the time domain, in accordance with an aspect of the invention.

[0037] FIG. 3 is a flow chart illustrating a method to construct a single-block physics- informed reduced-order learning model for simulations of electromagnetic / optical structures in the frequency domain, in accordance with an aspect of the invention.

[0038] FIG. 4 is a flow chart illustrating a method to construct a multi-block physics- informed reduced-order learning model for simulations of electromagnetic / optical structures in the frequency domain, in accordance with an aspect of the invention.

[0039] FIG. 5. A 2D photonic crystal used to illustrate the efficiency and accuracy of the method of the invention in the frequency domain

[0040] FIG. 6. POD eigenvalues in descending order at the r, M and X points in the Brillouin zone.

[0041] FIG. 7. Optical band structure calculated from the physics-informed learning (POD-Galerkin) model (solid lines) and DNSs (dot lines). The table includes the 8 frequency states and their errors at the M point predicted by POD-Galerkin.

[0042] FIG. 8. Profiles of |Ek|2in frequency states 1, 4, 7 and 8 in x andy directions through the peaks. Results predicted by POD-Galerkin are compared to DNS results that are denoted by black solid lines.

[0043] FIG. 9. Relative LSE for the 8 frequency states with respect to the DNS results at the M point.DETAILED DESCRIPTION OF THE INVENTION

[0044] The present disclosure describes a simulation method for electromagnetic and optical structures.

[0045] The methods of this invention provide an effective simulation tool that generates (i) space-dependent electric and magnetic fields and scalar and vector potentials in both time and frequency domains and (ii) electromagnetic / optical band structures in the Brillouin zone for periodic structures with or without defects or imperfections. The generated data for the fields, potentials and band structures can then be imported to electromagnetic / optical design tools to analyze physical properties / parameters of the designed structures, such as characteristic impedance, dispersion relations, reflection / transmission coefficient, S-parameters, etc. These properties / parameters are essential for designing electromagnetic / optical interconnects, waveguides, sensors, filters, antennas, etc. In addition, the simulation tool derived from the invention can be combined with other physics models or physics simulation tools to perform multiphysics simulation. These models / tools include commercial or open-source simulators, such as models / simulators included in Ansys HFSS, Ansys Lumerical, CST Studio Suite, COMSOL Multiphysics and OpenEM.. For example, Interacting with a heat transfer simulator, the simulation tool derived from the invention will be able to account for thermal influences on the above- mentioned properties / parameters of electromagnetic / optical devices and structures.

[0046] The invention is based on a projection concept that transforms the electromagnetic or optical problem from a physical space onto a reduced-dimensional space described by a finiteset of basis functions that are trained, optimized and generated by numerical solution (or training) data. The selection of the learning algorithm for training the basis functions to represent the physical problem of interest is not unique. Although the descriptions and examples for the methods of this invention are based on proper orthogonal decomposition (POD), together with Galerkin projection, this invention includes other learning methods for generating a set of optimal basis functions. The key objective of these physics-informed projection-based learning models is to transform the numerical problem of the electromagnetic wave equation with a large number of degrees of freedom (DoF) onto a reduced-dimensional space constituted by a small number of DoF. The methods offer efficient solutions for the electromagnetic fields with high accuracy while maintaining spatial resolution as fine as DNSs. For periodic structures with or without defects or imperfections, the frequency-domain physics-informed approaches also offer electromagnetic or optical band structures with extremely high accuracy and efficiency.

[0047] To achieve the above-mentioned objective and applicability for a wide range of applications, this invention includes two methods to construct the models for any electromagnetic / optical structures of interest, one in the time domain and the other in the frequency domain. Each method can be performed using either the global (i.e., single-block) approach or the multi-block approach. The single-block approach constructs the simulation model globally for the entire simulation structure, which is suitable for small structures. The multi-block approach, however, partitions the simulation structure into subdomains or blocks, which is more useful for large and complex structures, where global training of the entire structure becomes computationally prohibitive. Standard building blocks can be chosen as the subdomains for a category of technology. The trained basis functions (or modes, which represent the reduceddimensional space) and model parameters of standard or non-standard building blocks can be stored in a technology library (database) for constructing single-block or multi-block structures and to facilitate cost-effective engineering design / analysis of electromagnetic and optical structures.

[0048] Note that, in periodic structures with or without defects, dislocations or imperfections in the frequency domain, Bloch’s theorem (a well-known approach for wave propagation in the periodic structures) is applied to the electromagnetic wave equation using the Bloch function. This applies to DNSs for both single-block and multi-block approaches. The wave equation in the frequency domain hereafter is referred to as the electromagnetic wave equation interms of the periodic function in the Bloch function, where the periodic function has the same periodicity of the optical / electromagnetic lattice. Thus, the models for the method of the invention in the frequency domain are expressed in terms of the periodic function for periodic structures with or without the defects, dislocations or imperfection. The solution / data of electric field, magnetic field, scalar potential or vector potential obtained from DNS or the projection learning methods in the frequency domain thus represents the periodic function in the Bloch function.

[0049] As mentioned above, the electromagnetic wave equation can be represented by electric field, magnetic field or vector potential. When solving the wave equation for vector potential with the Lorenz gauge, magnetic field is evaluated from the curl of vector potential. If electric field solution is needed, solutions of the scalar-potential and vector-potential wave equations are also needed, and electric field is then calculated from scalar potential and vector potential. Formulations for the scalar-potential and vector-potential wave equations and expressions of electric and magnetic fields in terms of scalar potential and / or vector potential are well presented in nearly all graduate-level textbooks in electromagnetics. The procedure to transform the wave equation for electric field, magnetic field, scalar potential or vector potential to a reduced-dimensional space is the same although detailed formulations are different. The methods of the invention include models for electromagnetic wave propagation represented by the wave equation in terms of any of electric field, magnetic field, scalar potential and vector potential. However, the descriptions below only include the electric field representation for the TE wave and the magnetic representation for the TM wave.

[0050] Below the single-block approach is presented first for TE and TM waves in the time and frequency domains, followed by the multi-block approach.Time Domain Electromagnetic Wave Equation for Electric Field - TE Waves

[0051] The time-dependent electromagnetic wave equation for electric field E (r, t) is given as

[0052] For lossless media, electric conductivity <7 = 0, the wave equation becomes

[0053] Considering TE waves with the propagation direction along z and E = xEx, (2) is simplified to

[0054] Note that (3) can be written differently, for example, by dividing (3) by [io€, y.oor6. In each case, formulation of electromagnetic wave equation in the reduced dimensional space will be different.

[0055] Reduced-Dimensional Space - TE Waves

[0056] To project the problem onto reduced dimensional space, electric field is expressed in terms of a linear combination of the modes rjj (r) that describe the space,

[0057] Galerkin projection of (3) onto z / y (r), together with divergence theorem gives rise to,

[0058] Us al direction to thesurface S, a set of M ordinary differential equations (ODEs) can be derived. The A / -dimensional ODEs can be expressed as

[0059] where d =the matrices Ceand Geare determined by the modes, and the source vector Sein the reduced dimensional space includes the interior excitations and boundary conditions (BCs) given on the right-hand side of (5). Most coefficients or matrix / vector elements in (6) can be pre-calculated using the modesand stored in the database for simulation in the reduced-dimensional space to solve de(t). The coefficients of the second order derivative (i.e., elements of Ce) in (6) are expressed as

[0060] Expressions for the elements Geij and Seiin terms of the modes and sources are however dependent on the BCs.

[0061] Several types of BCs are given below to illustrate how to implement different BCs in the reduced dimensional space for Getj and Seiin (6). For each type of BCs, formulations are needed to appropriate include the boundary effects. dE

[0062] For a Neumann BC, = fs(rs, t) is applied, where fsis a given function (maybea constant or zero) on the boundary surface and rsindicates the location on the surface. In this case, the BC has no influence oGe

[0063] With this BC, the source vector elements are expressed as

[0064] During the simulation in the reduced dimensional space, the surface integral (a much smaller number of grids than a volume integral) can be estimated quickly. The volume integral in (9) usually involves very small volumes of source excitations (such as antennas), and can be evaluated efficiently as well.

[0065] For a Dirichlet BC with Ex= gs(rs, t), where gscould be any given function (may be a constant or zero) on the surface, several different approaches can be applied. Considering a simple approach by setting E-Jdn = h(Ex— gs~) on the boundary surface with h as a positive proportional constant. As h increases, Exapproaches gs. Applying a very large value of h, the boundary field Exwill become nearly the same as gs. However, if h is extremely large in this approach, numerical instability may occur. Using this approach with Exgiven in (4), the elements Geij become and

[0066] For a boundary with extremely high conductivity (such as metal), if Exis the tangential electric field on the boundary, setting gs= 0 with a large h value, Exbecome nearly zero.

[0067] In addition to Neumann and Dirichlet BCs, there are other different BCs for the electromagnetic wave equation, such as perfectly matched layers, periodic boundaries, scattering BCs, total field scattered field boundaries, etc. Note that, while performing the surface integrals over the entire domain, the BC on each surface or a portion of each surface may be different from others. A combination of different BCs is possible. Appropriate formulations in the reduced dimensional space for the surface integrals in (5) need to be carried out depending on the types of BCs in each surface area to obtain appropriate expressions for elements Gij and S[ in (6).Frequency Domain - TE Waves

[0068] The electromagnetic TE wave equation for electric field in the frequency domain can be derived from the time-dependent equation. For the propagation direction along z with d to

[0069] where to is the radial frequency, c is the speed of light and 6ris the relative permittivity. Electric field can be expressed in terms of a linear combination of the modes 77y(r):

[0070] Following the step-by-step procedure presented above in the time domain, a reduced-dimensional model in the frequency domain can be derived from (12) using (13).Periodic Structures - TE Waves

[0071] For periodic structures, such as photonic crystals, electromagnetic / optical superlattices, electromagnetic / optical metamaterials, etc., the wave equation for TE waves in (12) is rewritten below as an eigenvalue problem,

[0072] In periodic structures, Bloch theorem can be applied, and electric field for TE waves is expressed as a Bloch function,

[0073] where Ekis a periodic function with the same period as the structure and elk ris a modulating plane wave. Substituting (15) into (14), the wave equation for TE waves becomes

[0074] is the eigenfrequency of the periodic structure specified by the state k.Reduced-Dimensional Space - TE Waves

[0075] Using the projection-based approach for a periodic structure, Ek(r) can be expressed by a linear combination of the generate basic functions J / k(r),

[0076] where M is the selected number of basis functions (modes) to represent the solution, and ak nis the weighting coefficient for T / kn.

[0077] Similarly to the process to proj ect the time-domain wave equation onto the reduceddimensional space, Galerkin projection of (16) onto each of the modes f]k mis performed,

[0078] Using (17), (18) become a matrix eigenvalue equation,

[0079] Time Domain Electromagnetic Wave Equation for Magnetic Field - TM Waves

[0080] Because the permittivity is spatial dependent, the wave equation for magnetic field H(r, t) appears to be more complicated and is given as

[0081] In lossless media,

[0082] For TM waves with the propagation direction along z and H = xHx(thus J = yjy+ z / z), (23) becomes for plane wavesReduced-Dimensional Space - TM Waves

[0083] In the reduced-dimensional space, the magnetic field is expressed by a linear combination of the modes,

[0084] Following the same procedure for the TE waves, Galerkin projection of (24), together with (25), gives rise to

[0085] Or the wave equation in the Af-dimensional space can be written given as

[0086] where the source vector Shin the reduced dimensional space includes the interior excitations and BCs given on the right-hand side of (26). Most coefficients or matrix / vector elements in (27) can be pre-calculated using the modesand stored in the database forsimulation in the reduced order space to solve a.. The coefficients of the second order derivative (i.e., elements of Ch) in (27) are expressed as

[0087] Expressions of the elements Ghiand Shiin terms of the modes and sources are however dependent on the BCs. Derivations of the expressions for Ghiand Shican be performed similarly to the procedures described above for TE waves.Frequency Domain - TM Waves

[0088] The wave equation for TM waves in the frequency domain can be derived from the time-dependent equation. For the propagation direction along z with H(r, t) =

[0089] Magnetic field can be expressed in terms of a linear combination of the modes

[0090] Following the step-by-step procedure presented above in the time domain, a reduced dimensional model in the frequency domain can be derived from (29) using (30).

[0091] Periodic Structures - TM Waves

[0092] For periodic structures without sources, the wave equation for TM waves in (29) is rewritten as

[0093] In periodic structures, Bloch theorem can be applied, and magnetic field is expressed as a Bloch function,

[0094] where Hkis a periodic function with the same period as the photonic structure and elk ris a modulating plane wave. Substituting (32) into (31), the wave equation for TM waves becomesReduced dimensional space

[0095] In the reduced-dimensional space, Hk(r) can be expressed by a linear combination of the generate basic functions ?]k(r),

[0096] where M is the selected number of basis functions (modes) to represent the solution, and ak nis the weighting coefficient for r]k n. Similarly to the process to project the time-domain wave equation onto the reduced-dimensional space, Galerkin projection of (33) onto each of the generated modes r]k m, together with (34), is performed, which leads toMulti-Block Approaches

[0097] For the multi-block approach, BCs of each block are influenced by the adjoining blocks. The surface integrals in the above expressions in the reduced-dimensional space for the multi-block approach need to be reformulated to ensure interface continuity and numerical stability by using an interface smoothing technique. Although the discontinuous Galerkin method is presented below to illustrate the concept and procedure, the methods of the invention cover other smoothing techniques to reach appropriate and stable solutions at interfaces. The formulations of the electromagnetic wave equation in the reduced-dimensional space are briefly carried out only for TE waves. A similar step-by-step procedure can be performed for TM waves.Multi-blocks Time Domain - TE Waves

[0098] Electric field of TE waves for the / ?th block is expressed by a linear combination of the generate / rth-block modes,

[0099] where the j'th mode Tlpjtf) is generated from the electric field training data for the j>th block, apj (t) is its weighting coefficient, Mpis the number of modes in the j>th block. Galerkin projection of the time-dependent single-block electromagnetic wave equation for TE waves given in (5) is rewritten for the 72th block as

[0100] where the discontinuous Galerkin method [Arnold, etal. SIAM.J. Numer. Anal., vol.39, pp. 1749-1779, 2002] is applied to the surface integrals, and g is the penalty parameter defined as Np / dr with dr being the local mesh size at the interface and Npas the non-unit penalty number. Additionally, [*]PQand (*)PQare the difference and average operators, respectively, across the interface of the / rth and <yth blocks. Accuracy of the predicted solution may be influenced by the selected value of Np. Surface integrals in (40) vanish for the gth blocks with no interface with the j9th block. For a structure with a large number of blocks, the system becomes highly sparse. Similar to how (6) is derived for the single block model, using (39) in (40), a reduced (Mp)dimensional model for the pth block, represented by a set of Mpcoupled ODEs can be derived, accounting for effects from the neighboring blocks.Multi-Block Frequency Domains

[0101] In the frequency domain, formulations for the multi-block reduced-dimensional model are presented below only for periodic structures. For general cases in the frequency domain, following the same procedure presented in (39) and (4) for time-dependent multiblock model, electric field in the frequency domain given in (13) can be applied to Galerkin projection of the wave equation in (12) for the pth block to derive a multiblock reduced-order learning model.Multi-Block Periodic Structures

[0102] For periodic structures, Bloch theorem is applied. Similar to the formulations presented in (15) -(17) for the single block approach, the periodic function in the Bloch function for electric field in the pth block is given as modes with ap knas the weighting coefficients and Mpis thenumber of modes for the pth block. Galerkin projection of the single-block electromagnetic wave equation in the frequency domain for TE waves given in (18) is rewritten for the pth block as

[0103] Similar to (40), the discontinuous Galerkin method is applied to the surface integrals in (42), and Npneeds to be adjusted to obtain an accurate solution. The integrals vanish for the £ / th blocks with no interface with the pth block. For a structure with a large number of blocks, the system becomes highly sparse. Using (41) in (42), a reduced dimensional model forthe pth block represented by a A^-dimensional matrix eigenvalue equation can be derived, accounting for effects from the neighboring blocks.

[0104] For the multi-block approaches, the trained block models derived from (40) or (42), including their modes and model parameters in the reduced-dimensional space, are stored in a database. For a large structure, the database can be used to select the trained blocks to construct a M?-block structure with interface continuity between blocks enforced by interface coefficients derived from the discontinuous Galerkin method given in the surface integrals of (40) and (42). Note that the total number of modes (or DoF) for the entire A^-block structure M = Mi + M2 + ■■■ + Mp+ ■■■ + MNQ. If the field solution in the entire structure is needed, (39) or (41) needs to be post-processed over all NB blocks. Also note that in a multi-block structure the outer boundaries of the entire domain need to satisfy the BCs that were discussed above for the single-block approach.

[0105] Although the discontinuous Galerkin method is presented in (40) and (42), to illustrate the concept and procedure, other smoothing techniques can be applied as long as accurate and stable solutions at interfaces can be reached.

[0106] The methods of the invention are described step by step below based on the formulations and concepts presented above. The first method of the invention for the block-based projection learning simulation models is for electromagnetic simulation in the time domain and illustrated in FIG. 1 and FIG. 2 for single-block and multi-block approaches, respectively. There are three main components needed to construct the model for each approach, as described below. Firstly, training data are collected from DNSs to generate optimal basis functions (modes), based on but not limited to POD, neural networks, machine learning or artificial intelligence (Al), to represent the reduced-dimensional space. Secondly, Galerkin projection of the electromagnetic wave equation is performed to derive and close the reduced dimensional model. Lastly, the model parameters are precalculated, and the generated modes / eigenvalues and the model parameter are stored in a database that are then used to construct simulation models for single- or multi-block electromagnetic / optical structures.

[0107] In the first component, the generated basis functions need to be optimal and effective such that only a small number of modes are needed in simulation to reach a desired accuracy. To achieve this, a hybrid approach can be used to enhance training effectiveness and also to improve the model ability to account for nonlinear and anisotropic effects inelectromagnetic devices / structures. For example, when using POD, the generated / trained modes offer the best least squares fit to the training data. To generate an optimal and effective set of modes, training data must cover all possible parametric variations in the device / structure, and a massive amount of training data will be needed in devices / structures with complex parametric variations. In this case, grouping and clustering techniques, neural networks, machine learning or Al can be implemented in POD to further improve the training effectiveness and minimize the amount of training data needed to generate effective and optimal basis functions. For the second component, Galerkin projection presented in (5), (26) or (40) for TE or TM wave in single- or multi-block structures further incorporates physical principles that enhance the efficiency and accuracy of the models and empower their extrapolation ability. For the third component in the multi-block approach, the blocks to be trained and stored in the database should be primarily the “building blocks” for the technologies of interest, which will offer resources for cost effective design and analysis of electromagnetic / optical structures. However, some non-standard blocks will also be needed in many cases.

[0108] The model is constructed by first collecting numerical data of electric field or magnetic field at each or every several time steps from DNSs of the electromagnetic wave equation subjected to parametric variations for the entire structure when using the single-block approach given in Step 1 of FIG. 1 or for each block when using the multi-block approach given in Step 11 of FIG. 2. The parametric variations in the time domain may include, but not limited to, material properties and interior or / and boundary excitations of electromagnetic waves or current in time and / or in space at a fixed frequency or with a frequency spectrum. Using the collected training data, a finite set of basis functions is generated to establish a reduced-dimensional space, as described in Step 2 of FIG. 1 for the entire structure or Step 12 of FIG. 2 for the selected block in the multi-block approach. This for example can be achieved using the standard POD procedure via the method of snapshots [Sirovich, Quart. Appl. Math., vol. 45, pp. 561-571 and 573-590, 1987] where the POD eigenvalue of each POD mode represents the extracted information in the mode on the solution squared embedded in the training data. If the POD approach is used, the POD eigenvalue spectrum in terms of the POD mode index for the entire structure or for each block can then be used as a guideline to select the number of modes (or the DoF) needed in the prediction based on the desired accuracy.

[0109] The second component involves Galerkin projection of the electromagnetic wave equation onto the generated / trained modes. For the time domain, this leads to a set of ODEs in the reduced-dimensional space, as shown in Step 3 of FIG. 1 for the entire structure or in Step 13 of FIG. 2 for each block that includes interface couplings (influenced by the adjoining blocks) with interface continuities enforced by an interface smoothing technique, such as the discontinuous Galerkin method. The model parameters in the reduced dimensional space are then precalculated via the generated modes. These precalculated parameters include coefficients for these ODEs, excitations and BCs in the reduced dimensional space, as indicated in Step 4 of FIG. 1 for the single block approach. For the multi-block approach, as given in Step 14 of FIG. 2, the precalculated model parameters for each selected block in the reduced-dimensional space include coefficients of the ODEs, possible excitations and interface couplings induced by possible adjoining blocks derived from an interface smoothing technique. Additionally, the precalculated multi-block parameters also include coefficient of outer BCs for the blocks that may be exposed to any boundary in a multi-block domain. These precalculated model parameters, together with the trained modes and eigenvalues, for each block are stored in a database, as included in Step 5 of FIG. 1 and Step 15 of FIG. 2. These stored data, together with the ODEs in the reduceddimensional space, can be used for predictions / analysis of dynamic electromagnetic fields of the optical or electromagnetic structures. More modes (larger DoF) included in calculations offer a more accurate prediction but increase computational time.

[0110] To predict the solution of the dynamic electromagnetic equation in a domain, simulation in the reduced-dimensional space needs to be performed first, as shown in Step 6 of FIG. 1 for the single-block approach. After performing dynamic simulation of the structure in the reduced-dimensional space, post processing in Step 7 is carried out to calculate the dynamic field solution in real space. In post processing, the field solution is expressed as a linear combination of the modes. For TE waves in the time domain, electric field perpendicular to the propagation direction is given in (5). If the magnetic field is needed, Maxwell Faraday’s law can be applied to pre-calculate the curl of the modes and the magnetic field can be obtained from Maxwell Faraday’s law efficiently using the pre-calculated data. Similarly, for TM waves in the time domain, magnetic field perpendicular to the propagation direction is given in (25) expressed as a linear combination of the modes. If electric field is needed, Maxwell Ampere’s law can be applied to pre-calculate thecurl of the modes and the electric field can be obtained from Maxwell Ampere’s law efficiently using the pre-calculated data.

[0111] For the multi-block approach shown in Step 16 of FIG. 2 before performing dynamic simulation in the reduced-dimensional space in Step 17, the entire structure is constructed first by stitching the selected trained blocks whose modes and model parameters are stored in the database. This involves placing NB selected blocks and their model parameters together to establish NB sets of coupled ODEs for the entire As-block structure, where each set of ODEs represents one of the NB blocks. The interface continuity at each interface between adjoining blocks during simulation is enforced by their interface coefficients stored in the database. After the simulation (i.e., solving the NB sets of coupled ODEs) in the reduced-dimensional space, post processing in Step 18 is needed to calculate the field solution. This is similar to the single block approach; i.e., for TE waves the electric field perpendicular to the propagation direction is calculated for each block from a linear combination of the modes in (39). If the magnetic field is needed, similar to the single-block approach, Maxwell Faraday’s law can be applied to pre-calculate the curl of the modes and the magnetic field can be obtained from Maxwell Faraday’s law efficiently using the pre-calculated data. Although the dynamic reduced-order learning model for TM waves in multiblock structures was not derived above, its concepts and formulations are the TE wave model. For TM waves, magnetic field perpendicular to the propagation direction is calculated for each block from the expression below as a linear combination of the modes that are generated from training data of magnetic field,

[0112] If the electric field is needed, Maxwell Ampere’s law can be applied to precalculate the curl of the modes and the electric field can be obtained from Maxwell Ampere’s law efficiently using the pre-calculated data.

[0113] The second method of the invention for the block-based learning simulation models is for electromagnetic simulation in the frequency domain and illustrated in FIG. 3 and FIG. 4 for single-block and multi-block approaches, respectively. There are also three main components needed to construct the model for each approach, as described below.

[0114] Firstly, training data are collected from DNSs to generate optimal basis functions (for example using but not limited to POD, neural networks, machine learning or Al) to representthe reduced-dimensional space. Secondly, Galerkin projection of the electromagnetic wave equation is performed to derive and close the reduced dimensional model. Lastly, the model parameters are precalculated, and the generated modes / eigenvalues and the model parameter are stored in a database that are then used to construct simulation models for single- or multi-block electromagnetic / optical structures. If using POD in the first component, the generated / trained modes offer the best least squares fit to the training data. Therefore, the smallest number of DoF can be achieved in simulation to reach the desired accuracy if the training data quality is sufficient. Sufficient data quality requires accurate numerical data and thorough parametric variations are included in the DNSs for data collect. As described above for the dynamic model, a hybrid approach can be used to enhance training effectiveness and also to improve the model ability to account for nonlinear and anisotropic effects in electromagnetic devices / structures. For the second component, Galerkin projection presented in (18), (35) or (42) in the frequency domain for TE or TM wave in single- or multi-block structures.

[0115] Using the Bloch function, the reduced-dimensional model is constructed by first collecting numerical training data of electric field Ek(or magnetic field Wk) from simulations of (16) and (33) in a periodic structure over a selected number of frequency states at selected k points in the Brillouin zone. The training data are collected from DNSs of the electromagnetic wave equation subjected to parametric variations for the entire structure when using the single-block approach given in Step 21 of FIG. 3 or for each block when using the multi-block approach given in Step 31 of FIG. 4. The parametric variations in the frequency domain may include, but not limited to, material properties and periodic BCs. Using the collected training data, a reduceddimensional space is generated in terms of a finite set of basis functions at each selected k point for the single-block structure described in Step 22 of FIG. 3 or in Step 32 of FIG. 4 for the selected block in the multi-block approach. Similarly to the time domain method, one may generate the basis functions via, but not limited to, the POD procedure using the method of snapshots. The POD eigenvalue spectrum in terms of the POD mode index for the entire structure or for each block can then be used as a guideline to select the number of modes (or the DoF) needed in the prediction based on the desired accuracy.

[0116] The second component for the frequency-domain method of the invention utilizes Galerkin projection of the electromagnetic wave equation onto the generated / trained modes. This — > leads to an eigenvalue matrix equation in the reduced-dimensional space at each selected k pointfor the single-block structure shown in Step 23 of FIG. 3 or for each block shown in Step 33 of FIG. 4. For the multi-block approach in FIG. 4, this includes interface couplings (influenced by the adjoining blocks) with interface continuities enforced by an interface smoothing technique, such as the discontinuous Galkerin method. The model parameters in the reduced dimensional space are then precalculated via the generated modes. These precalculated parameters are the matrix elements in the eigenvalue matrix equation including boundary coefficients in the reduced dimensional space at each selected k point, as indicated in Step 24 of FIG. 3 for the single-block approach. For the multi-block approach, as given in Step 34 of FIG. 4, the precalculated model parameters for each selected block are the matrix elements in the eigenvalue matrix equation including coupling interface elements induced by possible adjoining blocks with interface continuities enforced by an interface smoothing technique. Additionally, the precalculated multiblock parameters also include coefficient of outer BCs for the blocks that may be exposed to any boundary in a multi-block domain. These precalculated model parameters, together with the trained basis functions and their eigenvalues are stored in a database at each selected k point, as included in Step 25 of FIG. 3 and Step 35 of FIG. 4. These stored data, together with the eigenvalue matrix equations in the reduced-dimensional space, can be used for predictions / analysis of electromagnetic fields of the optical or electromagnetic structures in the frequency domain.

[0117] To predict the solution of the electromagnetic equation in the frequency domain at each selected k point for the single-block approach, simulation in the reduced-dimensional space is performed first (i.e., solving the eigenvalue matrix equation), as shown in Step 26 of FIG. 3. This step also generates the band structure from the eigenvalues (i.e., eigenfrequencies) of the electromagnetic wave equation at selected k points in the Brillouin zone. After performing simulation of the structure, post processing is needed in Step 27 to calculate the field solution in physical space at each k point. Similar to the dynamic case, if the field solution is needed, electric or magnetic field for TE or TM waves, respectively, is solved first at each k point from the field expansion as a linear combination of the modes. Then Faraday’s or Ampere’s law is applied to calculate magnetic or electric field if needed.

[0118] For the multi-block approach in the frequency domain shown in Step 36 of FIG. 4 before performing simulation in the reduced-dimensional space at selected k points given in Step 37, the entire structure is constructed first by stitching the selected trained blocks whose modesand model parameters are stored in the database. This involves placing NB selected blocks and their model parameters together to form NB coupled set of eigenvalue equations in the reduceddimensional space at each k point. The interface continuity at each interface between adjoining blocks during simulation is enforced by their interface coupling elements stored in the database. This step also generates the band structure from the eigenvalues (i.e., eigenfrequencies) of the electromagnetic wave equation at selected k points in the Brillouin zone. After the simulation (i.e., solving the NB set of coupled eigenvalue equations), post processing in Step 38 is performed to calculate the spatial electric or magnetic field at selected k points if the field solution is needed. Similar to the dynamic case, after solving electric field for TE waves or magnetic field for TM waves from field expansion as a linear combination of the modes, Faraday’s or Ampere’s law is applied to calculate magnetic or electric field, respectively, if needed.

[0119] Calculations for DNSs (data collection), mode generation, model parameter evaluations, simulations of the physics-informed time-domain or frequency-domain models in the reduced dimensional space and post processing are performed using a combination of hardware and software. The calculations can be performed on computers via central processing units (CPUs) or graphics processing units (GPUs) with enough memory space and storage space. The memory space is usually provided by random access memory (RAM) for short-term data storage during computations. The storage space, such as hard disk drives (HDDs), solid-state drives (SSDs) or any other non-volatile storage devices, are usually for long term storage of output data. During calculations within each step of data collection, mode generation, model parameter calculations, simulation or post processing, numerical data need to be stored on RAMs temporarily, and the outputs of each step are stored in non-volatile storage devices for the next step calculations or later use.

[0120] The computer programs for the calculations can be on different operating systems, such as Linux or Windows. For DNSs in the training, any commercial software or open-source codes for electromagnetic or optical simulations can be used to collect training data. Popular simulators include Ansys HFSS, Ansys Lumerical, CST Studio Suite, COMSOL Multiphysics and OpenEM. For mode generation, model parameter evaluations, simulations of the physics-informed models and post processing, the programming language can be any programing language or one or a combination of the following: Python, Matlab, C, C++, Fortran, etc. For each of these languages, there are libraries that provide subroutines and functions to perform differentmathematical calculations, such as eigenvalue solvers for (19) and (36), ODE solvers for (6) and (27) and functions for numerical integrals for calculations of the model parameters for the eigenvalue equations and ODEs. For example, Matlab includes internal libraries for mathematical calculation, such as eigenvalue problems (e.g., eig and eigs) and ODE problems (e.g., ode45 for standard ODEs or ode89 for a higher-order solver with better accuracy). Popular libraries in Python include NumPy, Pandas and SciPy. Popular libraries for other languages include for example, LAP ACK, PETSc, Blaze, etc.

[0121] FIG. 5 illustrates an example of a 2D photonic crystal structure constructed by periodically repeated squares. This could be a structure for a photonic sensor or filter with a selected frequency window tuned by variation of dielectric materials inside the repeated squares. As shown in FIG. 5, each repeated square consists of 4 discs with diagonally symmetrical refractive indices with a refractive index n = 1 in the background. DNSs of the structure using the centra-difference method are applied to solve the electromagnetic TE wave equation given in (16) to collect the training data of for Ek(r). Parametric variations in training for this case include 20 samples of refractive indices randomly generated between 2 and 4 with diagonal symmetry, together with periodic BCs enforced in horizontal and vertical directions, which are implemented in DNSs to collect training data as a single block. In each sample, the training data for the first 10 frequency states are collected. There are thus 200 sets of data applied to generate the basis functions. In this case, POD is applied together with Galerkin projection (thus named POD- Galerkin), and one set of POD modes T]k n(r) and POD eigenvalues are generated for each k point in the 2D Brillouin zone. The first 65 POD eigenvalues at the T, M and A points in the Brillouin zone in descending order are given in FIG. 6. The first 10 POD eigenvalues decline slowly and drop rapidly beyond 10thmode. The model parameters given by (20) and (21) are evaluated using the generated modes f]kn(r). These parameters are coefficients of (19) resulting from Galerkin projection of (16). As stated in step 25 in FIG. 3 for the single-block in the frequency domain, the basis functions, eigenvalues and model parameters for each k point are then stored in a database, and the construction of the physics-informed learning model is thus completed for this photonic structure.

[0122] When performing simulation of the photonic structure, the selected values I if dielectric constants are shown in FIG. 5. As illustrated in step 26 in FIG. 3, the eigenvectors(whose element are ak n) and the eigenvalues are solved from the simulation of (19) in thereduced-dimensional space for each k point. The optical band structure for this photonic crystal can be evaluated in the Brillouin zone from tokand the post processing using (17) is performed to find electric field Ek(f). The outputs from the method of the invention for this demonstration are given below, together with their accuracy and efficiency. The outputs demonstrated below include optical band structure and electric field intensity. Depending on the needs of the design problems, the outputs of the physics informed models may include band structures and electric and magnetic fields, vector and scalar potentials in time and frequency domains. This example only demonstrates accuracy and efficiency of band structure and electric field in the frequency domain.

[0123] FIG. 7 illustrates the comparison of the optical band structures derived from the physics-informed learning model (in this case using POD-Galerkin) and the DNS using the centra- difference method. An excellent agreement is observed; their band structures overlap nearly perfectly with larger deviations near the Appoint. The table in FIG. 7 shows that errors at the M point for all 8 frequency states are below 0.048%.

[0124] Using the post processing in (17), Ekcan be solved. FIG. 8 displays [ZqJ2(that indicates the optical intensity) in space along x and y directions, solved from both the POD- Galerkin learning model and DNS for frequency states 1, 4, 7 and 8 via the peak intensity. Very good agreements are observed for all states using 5 modes except for State 8 that requires 8 modes. The least square error (LSE) of Ekderived from POD-Galerkin at the A / point of the Brillouin zone with respect to the DNS result is illustrated in FIG. 9 to further analyze the accuracy of the physics- informed projection learning model. The LSEs for States 1-7 are all near or below 3% when using 5 modes and below 2% when using 6 modes. The LSE for State 8 is somehow relatively high until 8 or more modes are used. The LSEs for all 8 states are near or below 1% when using 8 or more modes.

[0125] As to the efficiency in this demonstration, the physics-informed learning model offers a computational speedup over 10,000 time compared to DNS when only the band structure is needed. When the electric field spatial profiles at different frequency states are also needed in some photonic design for further calculations of optical properties, the speedup over the DNS is near 100 times because of additional computations needed in post processing. This is however for the single block model. In the multi-block cases for larger structures, DNS requires extremelyintensive computational resources due to much more numerical grid points (or DoF), the efficiency improvement for the methods of the invention over DNS will be higher. The computations for both the physics-informed learning model and DNS in this demonstration are carried out on an Intel il CPU with 8 cores in Matlab that performs parallel computations.

Claims

CLAIMSWhat is claimed is:

1. A computer-implemented block-based projection learning method for time-domain simulation of optical or electromagnetic devices, structure or system derived from physics-informed learning algorithms comprising the steps of: executing, by a processor of a computing system, direct numerical simulations of the dynamic electromagnetic wave equation for each selected block of the device, structure, or system; temporarily storing, in random access memory, numerical data generated during the direct numerical simulation for each selected block; storing, in non-volatile memory, training data derived from the direct numerical simulation for each selected block; generating, by the processor, basis functions and eigenvalues from the training data of each block to represent a reduced-dimensional space for the block, using a learning algorithm; performing Galerkin projection of the time-dependent electromagnetic wave equation for each selected block onto the reduced-dimensional space to construct a set of ordinary differential equations for the block in the reduced-dimensional space; enforcing the continuity for subdomain blocks at the interfaces between the selected block and variation of adjoining blocks by an interface smoothing technique; calculating, by the processor, model parameters for the ODEs for each selected block in the reduced-dimensional space including interface coupling coefficients influenced by variations of adjoining blocks resulting from the smoothing technique; and storing, in a database, the basis functions, eigenvalues and model parameters for each selected block.

2. The method of claim 1 further comprising constructing a single-block dynamic projection-based learning model using a selected block stored in the database for a single-block structure.

3. The method of claim 1 further comprising constructing a multi-block dynamic projection-based learning model using selected blocks stored in the database for a multi-block structure with interface continuities enforced by a smoothing technique.

4. The method of claim 1, wherein the electromagnetic wave equation can be expressed in terms of either electric, magnetic field, magnetic vector potential, or electric scalar potential, and optionally applying the Lorenz gauge during training data collection and Galerkin projection.

5. The method of claim 1, wherein the projection learning algorithm is derived from one or more of: reduced order learning methods, data grouping / clustering techniques, neural networks, machine learning or artificial intelligence.

6. The method of claim 1 wherein the block-based learning can be adapted to simulate devices, structures, systems and blocks with isotropic, anisotropic, uniform, nonuniform, linear and nonlinear materials.

7. The method of claim 1 further comprising performing simulation in the reduceddimensional space for a device, structure or system, that consists of one or more blocks with one or more basis functions for each block using the interface coefficients to ensure continuity at each interface between adjoining blocks with appropriate boundary coefficients for the entire structure.

8. The method of claim 7 further comprising performing post processing to calculate the spatiotemporal solutions for one or more electric field, magnetic field, scalar potential or vector potential, wherein the electric field is calculated from scalar and vector potentials when the vector potential and scalar potential are solved from their wave equations in their reduced dimensional spaces using Lorenz gauge.

9. The method of claim 8 further comprising using the post-processed results to calculate properties of the optical or electromagnetic devices, structures, or systems by coupling the projection-based learning models with other models or simulators to perform multi-physics simulations to further calculate properties of the said devices, structures, or systems, wherein the properties include at least one of the following: characteristic impedance, dispersion relations, reflection and transmission coefficient, and S-parameters.

10. The method of claim 1, wherein the magnetic field is calculated using pre-calculated curl data of electric-field modes based on Maxwell-Faraday’s law, and the electric field is calculated using precalculated data of magnetic field modes based on Maxwell- Ampere’s law.

11. The method of claim 1 , further comprising pre-calculating the curl of vector-potential modes and the gradient of scalar potential when solving the vector-potential and scalarpotential wave equations in their reduced dimensional spaces.

12. The method of claim 1 further comprising applying the said projection-based learning algorithm to consolidate smaller-size block models to generate a projection-based learning model for a larger-size block.

13. A computer-implemented block-based projection learning method for frequencydomain simulation of optical or electromagnetic devices, structure or system derived from physics-informed learning algorithms based on Bloch’s theorem comprising the steps of: executing, by a processor of a computing system, direct numerical simulations of the dynamic electromagnetic wave equation in the frequency domain for eachselected block at each selected k point in the Brillouin zone of the device, structure, or system; temporarily storing, in random access memory, numerical data generated during the direct numerical simulation for each selected block; storing, in non-volatile memory, training data derived from the direct numerical simulation for each selected block; generating, by the processor, basis functions and eigenvalues for each selected k point from the training data of each block to represent the reduced-dimensional space for the block, using a learning algorithm; performing Galerkin projection of the electromagnetic wave equation for each selected block in the frequency domain onto the reduced-dimensional space to construct an eigenvalue matrix equation in the reduced-dimensional space at each k point for the block; enforcing the continuity for subdomain blocks at the interfaces between the selected block and variation of adjoining blocks by an interface smoothing technique; calculating, by the processor, model parameters for the eigenvalue matrix equation for each selected block at each selected k point; and storing, in a database, the basis functions, eigenvalues and all model parameters at each k point for each selected block.

14. The method of claim 13, wherein a structure or subdomain is subjected to parametric variations at each selected k point in the Brillouin zone of the reciprocal lattice space for said structure or subdomain.

15. The method of claim 13, wherein the model parameters include interface coupling coefficients that are derived using an interface smoothing technique and account for variations in adjoining blocks.

16. The method of claim 13 further comprising constructing a single-block dynamic projection-based learning model in the frequency domain using a selected block stored in the database for a single-block structure.

17. The method of claim 13 further comprising constructing a multi-block dynamic projection-based learning model in the frequency domain using selected blocks stored in the database for a multi-block structure with interface continuities enforced by a smoothing technique.

18. The method of claim 13, wherein the electromagnetic wave equation can be expressed in terms of either electric, magnetic field, magnetic vector potential, or electric scalar potential, and optionally applying the Lorenz gauge during training data collection and Galerkin projection.

19. The method of claim 13 wherein the projection learning algorithm is derived from one or more of: reduced order learning methods, data-grouping / clustering techniques, neural networks, machine learning or artificial intelligence.

20. The method of claim 13 wherein the block-based learning can be adapted to simulate devices, structures, systems and blocks with isotropic, anisotropic, uniform, nonuniform, linear and nonlinear materials.

21. The method of Claim 13, further comprising generation of band structure in the Brillouin zone from the eigenfrequencies at each k point.

22. The method of claim 13 further comprising performing simulation of the eigenvalue matrix equation at each k point in the reduced-dimensional space to calculate eigenfrequencies and eigenvectors for a device, structure or system that consists of one or more blocks with one or more basis functions for each block using interface coefficients to ensure continuity at each interface between adjoining blocks with appropriate boundary coefficients for a structure.

23. The method of claim 22 further comprising performing post-processing using the eigenvectors to calculate on or more of: the spatial solution of electric field, magnetic field, scalar potential, or vector potential for each frequency state at selected k points, wherein the Lorenz gauge is used to calculate the spatial electric field from scalar and vector potentials at the selected k points.

24. The method of claim 23 further comprising coupling the said projection-based learning models in the frequency domain with other models or tools to perform multiphysics simulations to further calculate properties of the said devices, structures, or systems, wherein the properties include at least one of the following: characteristic impedance, dispersion relations, reflection and transmission coefficient, and S- parameters.

25. The method of claim 13 further comprising applying the said projection-based learning algorithm to consolidate smaller-size block models to generate a projection-based learning model for a larger-size block.

26. The method of claim 13, wherein the magnetic field is calculated using pre-calculated curl data of electric-field modes based on Maxwell-Faraday’s law, and the electric field is calculated using precalculated data of magnetic field modes based on Maxwell- Ampere’s law.

27. The method of claim 13, further comprising pre-calculating the curl of vector-potential modes and the gradient of scalar potential when solving the vector-potential and scalarpotential wave equations in their reduced dimensional spaces.

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