Ensemble-Based Global Search Framework for the Design Optimization of Fabrication-Constrained Freeform Devices

TL;DR

The paper tackles designing fabrication-constrained freeform devices with discrete materials by introducing Gaussian ensemble gradient descent (GEGD), which smooths a non-differentiable cost via Gaussian convolution and estimates gradients through ensemble sampling. By parameterizing the search with a mean latent density $\vec{\mu}_L$ and a Gaussian-embedded reward mean $\vec{\mu}_R$, GEGD achieves differentiability of the smoothed cost $f'$, while strictly maintaining feasible designs through ensemble sampling. The method leverages momentum-based updates, an RBF-based covariance, and control variates to greatly improve Monte Carlo efficiency, enabling effective exploration and convergence in high-dimensional spaces. Benchmark results on nanophotonic design problems show GEGD outperforms traditional gradient-based and population-based methods under equivalent compute budgets, demonstrating its potential as a general framework for density-based freeform design beyond nanophotonics. The approach is poised to enable efficient, fabrication-feasible optimization in domains where non-differentiable costs have previously blocked gradient-based methods.

Abstract

Although freeform devices with complex internal structures promise drastic increases in performance, the discreteness of the set of available materials presents challenges for gradient-based optimization necessary for the efficient exploration of the high-dimensional freeform parameter space. Several schemes have been devised to utilize a continuous latent parameter space that maps to actual discrete designs, but none thus far simultaneously achieves full differentiability and strictly feasible material choices during optimization. Here, we propose the Gaussian ensemble gradient descent framework, which transforms the piecewise-constant fabrication-constrained cost function by convolution with a Gaussian kernel to render it differentiable. The transformed cost and gradient are estimated through ensemble sampling, which is combined with variance reduction methods that greatly improve the sampling efficiency in high-dimensional parameter spaces. Furthermore, the use of ensemble sampling within a gradient descent framework leads to the effective hybridization of the exploration and exploitation strengths of population- and gradient-based methods, respectively.

Figures (6)

• Figure 1: A summary of freeform design parameterization schemes used under previous and currently proposed optimization frameworks. a, A direct parameterization of densities corresponding to feasible designs lead to a cost function with discrete domain and range, which presents difficulties for gradient descent. b,c, Latent parameters are used in more advanced frameworks such as the three-field method and the always-feasible parameterization to make the domain continuous. However, feasible designs only constitute a small fraction of the parameter space under the three-field parameterization (b), while the cost function is not differentiable everywhere under the always-feasible parameterization (c). d, Under the ensemble parameterization, the always-feasible cost function is probed using a Gaussian PDF whose mean is the optimization parameter. Every value for the mean $\mu_\text{R}$ represents a unique ensemble of feasible designs, and the objective is to minimize the average cost. Thus, both the domain and range of the optimization function can be made continuous without compromising design feasibility.
• Figure 2: Implementation details regarding the ensemble cost and gradient evaluations. a--e, Given a non-differentiable cost function (a), convolution with a Gaussian (b) yields a reparametrized cost function (equivalent to the expectation of the original cost sampled under the Gaussian distribution) that is differentiable with respect to the Gaussian distribution parameters (c). In practice, Monte Carlo sampling (d) is done on the original cost to estimate the expected cost (e). f, The ensemble gradient is computed by averaging unit vectors pointing from the mean towards sampled points, each scaled by the respective sampled cost and the distance of the sampled point from the mean. g, Contour plot of the expected cost showing the ensemble gradient computed from the samples in f.
• Figure 3: Methods for improving Monte Carlo sampling efficiency for freeform designs with large numbers of parameters. a, Examples of isotropic (top) and RBF (bottom) covariance matrices. The plots shown are $88 \times 88$ submatrices of the full $1260 \times 1260$ covariance matrices that produce the perturbations in d and f. b,c, An example mean reward matrix (b) and the corresponding feasible design (c). d,e, Isotropic perturbation (d) and subsequent feasible design generation (e) leads to a design that is only slightly different from the original design. f,g, RBF perturbation (f) results in significant topological changes in the feasible design (g). h--j, Plots describing the construction of low-variance gradient functions which are sampled to compute the ensemble gradient. Original ($f$) and control variate ($h$) cost functions (h) are multiplied by $g_\mu$ to construct the gradient function ($g_\mu f$) and its CV ($g_\mu h$) (i). Low-variance gradient functions are constructed by subtracting a scaled and shifted $g_\mu h$ from $g_\mu f$ (j). The approximate CV gradient function was computed using 10 high-fidelity and 100 low-fidelity samples ($r_\text{CV} = 10$). k, Monte Carlo sampling variance as a function of the low-to-high-fidelity sampling ratio ($r_\text{CV}$). The upper and lower limits represents variance for the original gradient function and the exact CV gradient function, respectively.
• Figure 4: Flowchart of the overall Gaussian ensemble gradient descent framework. The mean latent density is filtered and projected to yield the mean reward matrix. This is used to generate a reward matrix ensemble, each of which is fed into a feasible design generator. The cost for each feasible design is simulated and ensemble averaged to yield an estimate for the expected cost. The gradient of the expected cost with respect to the mean reward matrix is estimated using the same sampled costs. This gradient is backpropagated through the filtering and projection operations to yield the final gradient. The best sampled cost and the corresponding feasible design is saved and updated in each iteration and provided as the final output after a set number of iterations.
• Figure 5: Analytic test function benchmark results. a, A 2-dimensional cross section of the analytic cost function landscape through the origin along selected unit vectors $\hat{\rho}_{\text{L}1}$ and $\hat{\rho}_{\text{L}2}$. b, Cumulative best cost evolution during GEGD (left) with the corresponding feasible design at specific iterations (right). The algorithm performs topology exploration until approximately iteration 130, after which it fine tunes the shape of the obtained topology. c, The evolution of the Euclidean norm of the mean latent density during GEGD. d, Box-and-whisker plots of the best costs obtained using different optimization algorithms.