## 3 Accelerated Structural Topology Optimization
### The Deep Learning-Assisted Topology Optimization (DLTOP) Methodology
Considerable demands are posed even by moderately discretized domains, which becomes more pronounced in finer 3D meshes. As an example, the STO problem of a 3D bridge test case, discretized with a moderately dense mesh of 83,000 FEs, requires up to 7 h for performing 200 iterations of SIMP in serial CPU execution and up to 1 h in parallel GPU environment [4]. The DLTOP methodology can be applied to both serial or parallel, CPU or GPU execution implementations. DLTOP can be seen as a two-phase methodology. In the first phase, SIMP performs a limited number of iterations; the histories of the density values generated during these iterations are used as input data for the DBN. At the end of the first phase, DBN proposes an optimized distribution of the density values over the design domain based on this input data. Upon evaluation of the input data by the trained DBN, the latter performs a discrete jump from the pool of density values of the initial iterations to a close to final density for every FE. Subsequently, as part of the second phase, SIMP carries out a series of limited additional iterations, which corresponds to a fine-tuning process over the DBN-proposed distribution of optimized densities. A schematic representation of the two-phase DLTOP procedure to a single, randomly selected FE is depicted in Figure 2, where the abscissa correspond to the iterations performed by SIMP and the ordinate to the specific FE densities. Classification represents a challenging problem class for predictive models. Comparing the well-known regression predictive models with the classification ones, it is worth mentioning that the latter require information related to the complexity of a sequence dependence amid input parameters. For the STO problem, the density values of the early SIMP iterations represent the sequence dependence information that needs to feed the proposed (classification) methodology. The sequence of discrete-time data, i.e., the density value for every FE and the maximum number of iterations \(T\) required for convergence by SIMP, are generated by SIMP and stored in a density matrix \(D\), as shown below:  
\[D=\left[ \begin{array}{cccc}d_{1,1}&d_{1,2}&\ldots&d_{1,T}\\ d_{2,1}&d_{2,2}&\ldots&d_{2,T}\\ \vdots&\vdots&\ddots&\vdots\\ d_{ne,1}&d_{ne,2}&\ldots&d_{ne,T}\end{array} \right] \tag{11}\]  
A limited number of the iterations of the optimization procedure equal to the initial \(t\) iterations are used as time-series input data for training the DBN while the vector of the final SIMP iteration, i.e., the \(T\)th column of \(D\), is used as the target vector during the DBN training. \[\left[\begin{array}{cccc|c}d_{1,1}&d_{1,2}&\ldots&d_{1,t}\\ d_{2,1}&d_{2,2}&\ldots&d_{2,t}\\ \vdots&\vdots&\ddots&\vdots\\ d_{ne-1,1}&d_{ne-1,  
configuration after only a few SIMP iterations. This eliminates the greatest portion of the required SIMP iterations to obtain the final optimized structural topology, thus resulting in a substantial reduction in the required computing time and resources, while overcoming potential bottlenecks that would otherwise be encountered. Within the scope of further enhancing the computational efficiency of DLTOP, the authors envisioned a novel approach that combines deep learning with reduced-order modeling, which led to the development of the DLMU scheme presented in this paper. DLMU is based on DLTOP, however its novelty lies in the application of DLTOP to reduced-order models of the original system, instead of the system itself, and the subsequent extrapolation of their results to a full-scale optimized configuration of the original system, using the same deep learning techniques (DBNs). DLMU thus inherits the benefits of DLTOP and achieves equivalent accuracy, but substantially outperforms it in respect of computing time and demand in computing resources. In this section, the proposed DLMU scheme is described together with its implementation features.