Picard-Informed Basis Operators for Generalizable Few-Shot PDE Learning

Neural operators effectively solve parametric PDEs but are limited by large labeled data requirements and poor generalization under distribution shifts. We introduce Picard-Informed finite Basis Operators (PIBOs), a principled framework that enables robust generalization in the few-shot learning of parametric linear and nonlinear PDEs by leveraging finite basis representations and an unsupervised Picard-informed loss. For linear PDEs, PIBOs represent input functions using a compact finite basis, reducing learning complexity and enabling accurate operator learning from only a few training samples with robust generalization to arbitrary smooth inputs. For nonlinear PDEs, we exploit Picard iteration connections to linear operators and introduce an unsupervised Picard-informed loss that eliminates the need for paired input–output supervision. We provide generalization error estimates and demonstrate through extensive experiments that PIBOs achieve high accuracy and strong generalization with significantly reduced data requirements compared to existing neural operator approaches.