Data-driven methods and scientific machine learning are increasingly enabling new paradigms in computational solid mechanics, particularly for nonlinear boundary value problems where classical modeling and simulation approaches become prohibitively expensive or uncertain. Manufacturing processes such as deep drawing exemplify these challenges, combining high-dimensional field responses, strong nonlinearities, and limited observability of system-defining parameters. In this work, we present a data-driven framework that directly learns mappings between process parameters and mechanical field responses, bypassing explicit constitutive modeling and numerical solution of the governing equations [1]. The deep drawing problem is formulated as a mapping from input variables—material properties, tool geometries, and punch displacement—to high-dimensional output fields such as strain distributions. Convolutional neural networks (CNNs) are trained on 10,000 high-fidelity finite element simulations to learn this mapping. The resulting surrogate models accurately predict complex deformation patterns, including localized wrinkling, while providing orders-of-magnitude reductions in computational cost compared to conventional finite element solvers. This efficiency is particularly advantageous for design, optimization, and real-time evaluation of nonlinear mechanics problems. To address incomplete and uncertain inputs commonly encountered in experimental and industrial settings, we extend the deterministic surrogate modeling approach using a Diffusion Denoising Probabilistic (DDP) framework [2]. The probabilistic model learns distributions over admissible mechanical field responses conditioned on partial information, enabling uncertainty-aware predictions rather than single-point estimates. Finally, we outline an outlook toward inverse problems in sheet metal forming, where desired post-forming part or blank geometries are prescribed and corresponding tool geometries or process parameters are inferred. By combining learned forward surrogates with probabilistic generative models, the proposed approach establishes a foundation for data-driven inverse design and uncertainty-aware optimization in nonlinear solid mechanics.