Multi-Fidelity PDE Learning with Adaptive Spectral Neural Processes

Efficiently learning solution operators for parametric partial differential equations (PDEs) is difficult when high-fidelity simulations are scarce and expensive, even though low-fidelity approximations are often abundant. We introduce Multi-Fidelity Physics-Informed Neural Processes (MF-PINP), an adaptive spectral framework for multi-fidelity PDE learning that models each fidelity as an auto-regressive correction to the preceding level while imposing governing equations at resolution-dependent tolerances. Learnable spectral position encodings equipped with sparsity-inducing log-uniform priors select frequencies suited to each resolution and capture multi-scale solution structure. Physics-informed cross-fidelity attention then fuses observations according to spatial proximity and PDE residuals, allowing the model to discount unreliable coarse information. We derive PAC-Bayes generalization bounds connecting spectral complexity to model capacity and calibration bounds for uncertainty propagated through the fidelity hierarchy. Experiments across benchmark parametric PDEs show that MF-PINP improves predictive accuracy, maintains well-calibrated uncertainty, and substantially reduces the need for expensive high-fidelity observations.