KAPI-ONet: Kernel-Adaptive Physics-Informed Operator Network for PDEs

Physics-informed operator learning enables rapid prediction across families of PDEs without requiring paired numerical solution fields, but existing methods often rely on latent or globally shared representations that are difficult to interpret physically and can be computationally expensive to train. This raises the question of whether an operator learner can expose how variations in the input reorganize the solution representation while remaining compact enough to reduce the computational burden of physics-informed training. To address this gap, we propose KAPI-ONet, a kernel-adaptive physics-informed operator network in which each solution is represented explicitly by a task-dependent collection of Gaussian RBFs with interpretable centers, scales, orientations, and coefficients in physical or space–time coordinates. For functional inputs, KAPI-ONet formulates operator learning as a mapping between structured Gaussian-splat representations of the input and solution, providing a common coordinate-aware description of both sides of the operator through geometrically meaningful amplitudes, locations, and scales. Across elliptic, time-dependent, and Navier–Stokes problems, the learned kernel geometry adapts to localized and high-gradient solution structures, while KAPI-ONet attains competitive accuracy with substantially lighter training in settings where direct computational comparisons are available. We also evaluate KAPI-ONet beyond standard forward PDE operator learning, including parameter-to-field prediction and two biomedical applications: inverse hemodynamics from sparse velocity measurements and weakly supervised cardiac annotation. These results support task-adaptive kernel geometry as an interpretable and computationally efficient representation for operator learning across diverse problem formulations. Code and reproducibility materials are available at: https://anonymous.4open.science/status/kapi-onet-anonymous-6C97