Geometry Conditioning in Neural Operators through Representation Alignment

Neural operators provide efficient surrogates for physical simulation by learning mappings from problem specifications to solution fields. However, effectively incorporating geometric information into these models often relies on specialized architectural mechanisms, tightly coupling geometry handling with the underlying operator design. In this work, we explore a different perspective: geometry can guide operator learning independently of the operator architecture. We introduce GEometry-aware Alignment of Representations (GEAR), a lightweight regularization framework that treats geometry as a relational training prior. Rather than matching representations to predefined geometric embeddings, GEAR aligns their local relations with those defined by geometry, enabling lightweight geometric guidance directly during training. To balance geometric guidance with the physical prediction objective, GEAR suppresses alignment updates that conflict with the physical prediction objective, allowing geometric structure to guide learning without over-constraining task-specific features. This decouples geometry-aware learning from architectural design and enables the same training strategy to be applied across neural operator backbones. Experiments across multiple operator architectures and simulation datasets involving varying geometries demonstrate consistent improvements in predictive performance.