Physics-driven design seeks high-performing geometries, which need not conform to a learned shape category. Parametric and latent-space approaches facilitate surrogate-based optimization, but restrict deformations to prescribed shape families or learned decoders. We introduce the Mesh-to-Shell Neural Operator (M2S-NO) as the mesh-to-shell component of a category-agnostic inverse shape optimization framework. We separate the explicit mesh used for design from the volumetric representation used for physics prediction. M2S-NO maps oriented surface samples to a surface-centered shell representation, such as TruncatedSDF or HeatSDF. A separate problem-specific PDE neural operator (PDE-NO) maps this representation to physical fields. Objective gradients update mesh vertices through both operators, without a latent shape decoder. Preconditioning and remeshing support free-form deformation, while fixed interfaces are controlled directly on the mesh. Optional shell-space proposals enable topology changes. We evaluate geometric matching, jet-engine brackets, and neutron moderators, with independent FEM and OpenMC evaluation. Relative to the best training designs, M2S-NO reduces maximum bracket displacement by 14.3% and increases neutron-sensor current by 13.8%. Optimization also succeeds from an out-of-distribution bracket initialization, demonstrating the framework's geometric flexibility.

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