LINK: Learning Physical Interaction Kernels for Neural Operators

Neural operators on irregular domains require interactions that remain meaningful as discretization and geometry change. Attention provides a flexible mechanism for exchanging information between sampled locations, but its routing is typically determined through generic query–key interactions rather than directly from physical structure. We introduce LINK (Learned Interaction Kernels), a kernel attention mechanism that learns physical-space routing between locations using their coordinates, relative geometry, and the current problem instance. The resulting input-conditioned kernel integrates naturally into neural operator architectures while retaining flexible point-set computation. We characterize the resulting operator from continuum and representation perspectives, and develop scalable realizations for large discretizations. Across several PDE learning settings, LINK improves predictive accuracy and optimization, with the largest gains arising when interactions are nonlocal. These results show that learning routing directly in physical space provides an effective inductive structure for neural operators on irregular domains.