For photonic applications, the inverse design method plays a critical role in the optimized design of photonic devices. According to its two ingredients, inverse design in photonics can be improved from two aspects: to find solutions to Maxwell’s equations more efficiently and to employ a more suitable optimization scheme. Various optimization algorithms have been employed to handle the optimization: the adjoint method (AM) has become the one of the most widely utilized ones because of its low computational cost. With the rapid development of deep learning (DL) in recent years, inverse design has also benefited from DL algorithms, leading to a new pattern of photon inverse design. Unlike the AM, DL can be an efficient solver of Maxwell’s equations, as well as a nice optimizer, or even both, in inverse design. In this review, we discuss the development of the AM and DL algorithms in inverse design, and the advancements, advantages, and disadvantages of the AM and DL algorithms in photon inverse design.

Deep learning (DL) equipped iterators are developed to accelerate the iterative solution of electromagnetic scattering problems. In proposed iterators, DL blocks consisting of U-nets are employed to replace the nonlinear process of the traditional iterators, i.e., the conjugate gradient (CG) method and generalized minimal residual (GMRES) method. New implementations of the complex-valued batch normalization in the U-net are proposed and investigated in terms of the DL equipped iterators. Numerical results show that the DL equipped iterators outperforms their traditional counterparts in terms of computational time under the comparable accuracy since the phase information of the currents, fields and permittivity are properly handled.

The phase information's role in deep neural networks (DNNs) to solve the electromagnetic inverse scattering problems is investigated. The feedforward neutral network model with complex-valued (CV) data stream and the corresponding CV backpropagation training algorithm are proposed to realize CV convolutional neural networks. Numerical examples are carried out to demonstrate the phase information's role in DNNs in terms of generalization capability as well as the convergence speed in the training stage.

Unlike the universal approximation theorems for functions mapping from a real-valued (RV) vector to a RV number or from a complex-valued (CV) vector to a CV number, in the field of electromagnetism, we need to approximate functions mapping from a RV vector to a CV number when we consider the electric field as a function of the spatial coordinate in the frequency domain. Typically, CV numbers contain phase information. When such phase information is handled properly, the performance of the neural networks (NNs) can be improved. This work derives a universal approximation theorem for functions mapping from a RV vector to a CV number. A deep NN, named as HV-DL, is designed accordingly, which consists of a RV input layer, a RV module containing two branches, a CV module and a CV output layer. The proposed universal approximation theorem is verified by numerical experiments on the HV-DL solution of the two-dimensional (2-D) electric field integral equation (EFIE). To integrate the underlying physics of electromagnetic scattering into the proposed HV-DL, the loss function is computed according to the EFIE.

For applications with many right-hand sides (RHSs), to avoid the solution of m different RHSs repeatedly, previous work has employed the interpolative decomposition (ID) to figure out skeleton RHSs by exploiting the rank deficient property of the n ×m RHS matrix B for a system with n unknowns. However, the peak memory requirement for the skeletonization may degrade the performance of the previously developed algorithms when n becomes very large. To alleviate the associated difficulty, a strategy is proposed to construct a submatrix of B to figure out skeleton RHSs. Numerical experiments on different applications show the accuracy and efficiency of the proposed algorithms.