  A Fourier neural operator (FNO) is one of the physics-inspired machine
learning methods. In particular, it is a neural operator. In recent times,
several types of neural operators have been developed, e.g., deep operator
networks, Graph neural operator (GNO), and Multiwavelet-based operator (MWTO).
Compared with other models, the FNO is computationally efficient and can learn
nonlinear operators between function spaces independent of a certain finite
basis. In this study, we investigated the bounding of the Rademacher complexity
of the FNO based on specific group norms. Using capacity based on these norms,
we bound the generalization error of the model. In addition, we investigated
the correlation between the empirical generalization error and the proposed
capacity of FNO. From the perspective of our result, we inferred that the type
of group norms determines the information about the weights and architecture of
the FNO model stored in the capacity. And then, we confirmed these inferences
through experiments. Based on this fact, we gained insight into the impact of
the number of modes used in the FNO model on the generalization error. And we
got experimental results that followed our insights.
