Part I: Study of fracture properties of graphene--like two--dimensional materials using molecular dynamics (MD) simulationsGraphene is a monoatomic thick sheet of sp2-hybridized carbon atoms tightly packed in a honeycomb lattice structure. It has been studied for basic science and commercial applications due to its extraordinary thermal, optical, and mechanical properties. Since its discovery, it has drawn extensive attention to the science community for its unique 2D structure. In this research, we employed molecular dynamics simulations and machine learning methods to study mechanical and fracture properties of graphene--like two--dimensional materials (e.g., C3N, bicrystalline graphene, and polycrystalline graphene). Molecular dynamics simulations are used to study the mechanical and fracture properties of C3N. The impact of initial crack orientation on the crack path is studied by applying tensile strain to C3N sheets containing initial cracks in the armchair and zigzag directions. The results show that the cracks grow by creating new surfaces in the zigzag direction. The capability of Griffith theory and quantized fracture mechanics (QFM) in predicting the fracture strength of C3N is studied. Griffith theory can not predict the fracture strength of C3N if the crack length is shorter than 9 nm, while QFM shows better results at nanoscale cracks of C3N. The notch effects on the fracture strength of C3N are studied, and it is shown that notch effects are important in predicting the fracture strength of C3N. MD simulations predict a critical energy release rate of 10.982 J/m$^2$ for \ch{C3N} using the Rivling--Thomas method.Graphene sheets produced by chemical vapor deposition (CVD) are polycrystalline, and the presence of grain boundaries (GBs) alters their mechanical properties relative to single-crystal graphene. In this Study, we performed MD simulations using REBO2+S to establish a relation between the work of separation (fracture energy of grain GBs) and the misorientation angle of symmetric grain boundaries of graphene. The traction--separation laws (TSLs) of grain boundaries are extracted by using cohesive zone volume elements (CZVEs) ahead of the crack tip. The maximum traction and maximum separation distance values obtained for the CZVE predict that the TSLs of grain boundaries have a bilinear form. The areas under the traction-separation curves are used to calculate the work of separation (fracture energy) of the grain boundaries. The results show that as the grain boundary misorientation angle increases, the separation energy of the grain boundaries decreases.Finally, a deep convolutional neural network model has been developed to predict the mechanical and grain properties of polycrystalline graphene. The goal is to train the network such that it can predict Young's modulus and fracture stress of graphene sheets by analyzing an image of the polycrystalline sheet. The data required for training our machine learning model is generated using molecular dynamics simulations by modeling the behavior of polycrystalline graphene under uniaxial tensile loading. More than 2000 data points are generated for graphene sheets of different grain sizes and grain orientations. Part II: Study of topology optimization through neural networks for coupled thermo--mechanical problem.Topology optimization (TO) is a powerful computational design method for automatically generating a structural layout to determine the optimal material layout in a design domain with maximum performance under relevant design specifications set by the user. Conventional Topology optimization uses the finite element method to evaluate the design performance against defined criteria. This Study uses a density-based topology optimization method using neural networks (NN) to design domains composed of multi--materials under coupled thermo--mechanical loading. We investigate the application of NN as an optimization technique to conduct topology optimization. The proposed method relies on the popular density-based solid isotropic material with penalization (SIMP) mathematical formulation but uses a neural network's activation functions to represent the SIMP density. The neural network produces the topology density field by minimizing a loss function defined for the problem. A Fourier space projection has been implemented within the deep neural network model to control the minimum, and maximum length scales to meet the manufacturing and other functional requirements. We have adapted the high-performance Google automatic differentiation (AD) library JAX to build an end--to--end differentiable network for the machine learning model. The sensitivity computations are automated using the built--in backpropagation functionality in JAX. The performance of the proposed framework is demonstrated by solving several coupled thermo--mechanical compliance minimization problems for design domains composed of a single material or multi--materials and comparing the results with the optimized structures obtained from other topology optimization techniques.