  Fourier Neural Operator (FNO) is a popular operator learning framework. It
not only achieves the state-of-the-art performance in many tasks, but also is
efficient in training and prediction. However, collecting training data for the
FNO can be a costly bottleneck in practice, because it often demands expensive
physical simulations. To overcome this problem, we propose Multi-Resolution
Active learning of FNO (MRA-FNO), which can dynamically select the input
functions and resolutions to lower the data cost as much as possible while
optimizing the learning efficiency. Specifically, we propose a probabilistic
multi-resolution FNO and use ensemble Monte-Carlo to develop an effective
posterior inference algorithm. To conduct active learning, we maximize a
utility-cost ratio as the acquisition function to acquire new examples and
resolutions at each step. We use moment matching and the matrix determinant
lemma to enable tractable, efficient utility computation. Furthermore, we
develop a cost annealing framework to avoid over-penalizing high-resolution
queries at the early stage. The over-penalization is severe when the cost
difference is significant between the resolutions, which renders active
learning often stuck at low-resolution queries and inferior performance. Our
method overcomes this problem and applies to general multi-fidelity active
learning and optimization problems. We have shown the advantage of our method
in several benchmark operator learning tasks. The code is available at
https://github.com/shib0li/MRA-FNO.
