# Design Optimization of Truss Structures Using a Graph Neural Network-Based Surrogate Model
## 1 Introduction
In the studies that have focused on predicting optimal structural designs, the networks solve size optimization problems and directly predict the optimal cross-sectional areas of the structural members. For instance, [PERSON] et al. [24] developed a Deep Neural Network (DNN) to predict the optimal design of double-layer grids with pin joints under a defined loading scenario. In another study, [PERSON] et al. [25] trained a neural network to predict the optimal cross-sectional area of each bar and the minimum volume of a simple 3-bar truss under different loading conditions. Recently, [PERSON] and [PERSON] [26] presented a method to optimally design a 10-bar planar truss using a deep neural network. The DNN returns the optimal cross-sectional areas of the truss elements for different cases of applied loads and allowable stresses and displacements. [PERSON] et al. [27] proposed a novel DNN-based model by which a simultaneous truss shape and size optimization problem is decomposed into two simpler problems: a size optimization problem nested within a shape optimization problem. The DNN was trained to directly approximate the optimal cross-sectional areas of the members of a truss with a given shape configuration. Surrogate-assisted optimization has become increasingly popular. Surrogate models are prediction tools that can replace objective or constraint functions to accelerate the optimization process. [PERSON] and [PERSON] [28; 29] were among the first researchers who applied neural networks in the optimization process as a replacement for structural analysis. [PERSON] et al. [30] implemented a neural network-based surrogate model for size optimization of large-scale 3D trusses to approximate their structural responses during the optimization process. A similar method involving neural networks and genetic algorithm as the metaheuristic algorithm was proposed by [PERSON] et al. [31] for structural system reliability optimization. [PERSON] et al. [32] developed a data-driven framework and bypassed the use of constitutive models to improve the speed of topology optimization processes. In a similar vein, [PERSON] and [PERSON] [33] trained two neural networks to separately approximate the nodal displacements and the member axial forces for evaluation of the constraints in truss size optimization problems. [PERSON] et al. [34] proposed a neural network-based surrogate model for the optimization of truss structures. In their method, a DNN uses cross-sectional areasof the truss members as input and approximates the nodal displacements by including nonlinear effects due to large deformations. While traditional ANNs are primarily designed to process data that can be represented as feature vectors, Graph Neural Networks (GNNs) are specifically tailored to operate on graph-structured data. GNNs capture both the relational information between nodes and the features associated with each node within a graph, allowing them to learn complex interactions in the graph data. Although extensively explored for over a decade [35; 36; 37], GNNs' popularity has grown only in recent years. Several comprehensive reviews on graph neural networks have recently been published [38; 39; 40; 41; 42]. These networks have been effectively employed in various domains such as computer vision, molecular chemistry, molecular biology, pattern recognition, and data mining, in which the data relationships can be represented as graphs. For example, [PERSON] et al. [43] and [PERSON] et al. [44] represented chemical molecules as graphs and predicted their chemical properties using GNNs. [PERSON] et al. [45] implemented a GNN to learn accurate physical simulations, infer abstract properties of physical systems, and predict the physical trajectories of dozens of objects over any number of time steps. [PERSON] et al. [46] developed a mesh-based machine learning model for the prediction of deformation, stress, and strain fields in material and structural systems using a GNN model.