Physics-Informed Active Learning via Functional Simulated Annealing for Neural Operator

Neural operators—especially Physics-Informed Neural Operators (PINO)—learn solution maps for parametric Partial Differential Equations (PDEs), but their implementation is limited by the cost of generating high-resolution training data. We consider the common scenario in which the parameter space is known, but only coarse, low-fidelity solutions are available. To make the most of a limited budget, we introduce Physics-Informed Active Learning (PIAL). PIAL iteratively chooses the most informative high-resolution instances without ground-truth solutions by estimating how strongly each candidate violates the governing PDE when evaluated by the current operator. A simulated-annealing search maximizes this uncertainty criterion, steering computation toward samples that provide the greatest improvement. The result is a data-efficient workflow that accelerates PINO convergence while overcoming strict resource constraints.