Parametric Model Embedding (PME) [1] is an established design-space dimensionality-reduction technique [2] capable of capturing the geometric variance of parametrized configurations into a low-dimensional latent space. While its effectiveness has been proven across various engineering domains [3,4], its formulation, rooted in generalized eigenproblems, is inherently linear, limiting its ability to represent complex, nonlinear correlations between shape features and design parameters. This work proposes a nonlinear evolution of PME through a supervised deep learning framework designed to ensure geometric consistency. The architecture establishes a "round-trip" mapping: an encoder projects high-dimensional shape coordinates into a reduced latent manifold, and a decoder reconstructs the original design variables. Crucially, to overcome the limitations of standard loss functions that treat all design variables with equal statistical weight regardless of their impact on the shape, we introduce a differentiable geometric surrogate. This surrogate acts as a proxy for the shape generation process, allowing the integration of a geometry-consistent loss directly into the training loop. This forces the network to minimize the reconstruction error not only in the parameter space but also in the coordinate space, while strictly adhering to the design variable bounds through a dedicated boundary loss. A proof-of-concept application to a parametric hydrodynamic configuration demonstrates that the proposed nonlinear formulation achieves the same geometric variance recovery as linear PME with a lower latent dimensionality, resulting in more compact and efficient optimization spaces. The latent coordinates are not intended to replace physical design parameters, but to serve as abstract control variables that guarantee admissible and geometry-consistent configurations, thereby defining a well-conditioned reduced space for simulation-based design optimization.