A physics-informed geometry-embedded graph autoencoder for shape reduction, reconstruction, and surrogate modeling

High-dimensional shapes need reduced representations that are compact, reconstructable, mesh-aware, and predictive of physical performance. This is challenging because low-dimensional shape families may form nonlinear manifolds in nodal-coordinate space, so linear bases can require more modes than the number of independent deformation coordinates. This paper presents a physics-informed geometry-embedded graph autoencoder (PI-GAE). It encodes mesh or profile deformations into a shared latent vector, reconstructs coordinates through a singular-value-decomposition-assisted decoder, and predicts scalar or aligned field responses with condition-aware heads. Controlled translation-manifold studies compare the nonlinear latent requirement with proper orthogonal decomposition (POD) at matched reconstruction accuracy. The main transition follows the prescribed intrinsic dimension, although finite model capacity limits accuracy as intrinsic dimension or translation range grows. The method is evaluated on AirfRANS, OptiWing3D, SuperWing, and Rotor37, spanning two-dimensional airfoils, three-dimensional wings, and a turbomachinery blade surface with structured and neighborhood graphs. Held-out scalar prediction reaches coefficient-of-determination values from 0.946 to 0.992, and AirfRANS also demonstrates node-wise pressure-coefficient prediction. A parameter-matched comparison shows graph-based encoding matches simpler encoders on most cases but yields the clearest advantage on the highest-resolution SuperWing geometry. Mean normalized reconstruction error remains below 0.3% for AirfRANS, OptiWing3D, and Rotor37; the upper-deformation SuperWing quartile reaches 1.84%. The shared latent representation provides a compact, mesh-aware coordinate for both geometry reconstruction and physical-response prediction within a single model.