Fig. 3. Setup and results of a compliance minimization problem with 11 × 11 design variables.

a Problem setup: minimizing compliance subject to 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, CMA-ES denotes Covariance Matrix Adaptation Evolution Strategy. 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. $\tilde{E}=0.222$. e Optimized design produced by SOLO. n_train = 10,243 and $\tilde{E}=0.228$. f Optimized design produced by SOLO. n_train = 77,691 and $\tilde{E}=0.222$. In d–f dark red denotes ρ = 1 and dark blue denotes ρ = 0, as indicated by the right bar.