Topology optimization has gained significant attention since its inception a few decades ago, producing theoretically and mathematically suitable optimized results. However, their industrial application has been limited. Recent advancements, particularly in additive manufacturing, have renewed interest by enabling the realization of such optimized results. Furthermore, the increasing development experienced by numerical analysis in terms of complexity, driven by the rise of artificial intelligence and scientific machine learning offer more efficient ways to perform topology optimization. This allows its further use in fields such as mechanical metamaterials, multiscale optimization or inverse problems. For instance, dimensionality reduction techniques via the use of deep learning have enabled the creation of surrogates to compute fast forward evaluations, reducing computational time. These approximations of complex physical model e.g., finite elements (FE) and optimization, allow for more efficient solutions to computational challenges, such as inverse problems. This paper presents a novel methodology using Rank Reduction Autoencoders (RRAEs) [1], for efficient predictions in both forward and inverse solid mechanics problems. This approach bypasses the potentially expensive forward problem (geometry to solution) and addresses the ill-posed inverse problem (solution to geometry). The methodology involves the creation of three models: two RRAE models generate low-rank approximations of both the geometries and solutions, while a third model relates the latent coefficients of both. The latter can be trained from geometry latent coefficients to solution latent coefficients (direct problem) and vice versa (inverse problem). Lastly, these models not only enable efficient computations for direct and inverse problems, but also serve as a generative design tool through interpolations in the latent space to generate new designs. These capabilities hold potential for parameterized topology optimization and, ultimately, for more generic multi-physics problems.