Xinyi Li, Xi Deng, Yubo Cao, Hudson Liu, Anima Anandkumar
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.
Predicting molecular magnetic properties often requires costly electronic-structure calculations that account for spin-orbit coupling. Inspired by the spinor nature of the underlying electronic states, we introduce eSpinor, an SU(2)-equivariant graph neural network that propagates spinor features alongside the integer-rank tensors used in conventional equivariant models. We initialize spinors from eigenvectors of an approximate Hamiltonian that includes spin-orbit coupling. The network preserves their rotation laws and makes predictions independent of arbitrary eigenbasis choices. Comparisons with geometry-only and Hamiltonian-informed baselines assess the benefit of this architecture beyond supplying electronic information. On cobalt single-molecule magnet candidates, eSpinor improves magnetic anisotropy predictions, including for unseen ligands. It also improves excitation-energy predictions for dysprosium candidates with unseen ligands, extending the benefit to rotationally invariant scalar targets. For a dysprosium magnetic resonance imaging probe, it predicts real electron-spin correlations and low-field nuclear relaxation rates more accurately despite static crystal-field errors similar to the Hamiltonian-informed baseline. This dynamical advantage extends from water to methanol without retraining. These results support spinor message passing as a useful inductive bias for molecular magnetism in the tested systems and show why model selection should be guided by accuracy on the intended physical observables.