Fig. 2. Setup and results of a compliance minimization problem with 5 × 5 design variables.

a Problem setup: minimizing compliance subject to a maximum volume constraint. b Best dimensionless energy with a total of n_train accumulated training samples. SOLO denotes our proposed method where the cross “X” denotes the convergence point (presented in e), “Offline” denotes training a DNN offline and then uses GSA to search for the optimum without updating the DNN, whose results are independent so they are plotted as circles instead of a curve, SS denotes Stochastic Search, which is the same as SOLO except that $\widehat{\mathit{\rho }}$ in each loop is obtained by the minimum of existing samples, CMA-ES denotes Covariance Matrix Adaptation Evolution Strategy, BO denotes Bayesian Optimization. SOLO converges the fastest among these methods. c Energy prediction error of $\widehat{\mathit{\rho }}$ relative to FEM calculation of the same material distribution. e($\widehat{\mathit{\rho }}$) denotes DNN’s prediction, E($\widehat{\mathit{\rho }}$) denotes FEM’s result. d Optimized design produced by the gradient-based method. $\tilde{E}=0.293$. e Optimized design produced by SOLO. n_train = 501 and $\tilde{E}=0.298$. f Optimized design produced by SOLO. n_train = 5782 and $\tilde{E}=0.293$. In d–f dark red denotes ρ = 1 and dark blue denotes ρ = 0, as indicated by the right color scale bar.