Refinement Does Not Amplify Discretization Mismatch in Fourier Neural Operators

Neural operators provide a powerful framework for learning mappings between function spaces, including solution operators for partial differential equations (PDEs). Fourier neural operators (FNOs) are particularly attractive because their parameters are independent of the spatial resolution, enabling a model trained on one discretization to be evaluated on another. However, resolution-independent parameters do not imply discretization-invariant predictions: the outputs of an FNO can differ when the same input function is represented on different discretizations. This discretization mismatch is important for assessing the reliability of FNOs across discretizations, yet a precise theoretical characterization of how the output discrepancy depends on the two discretization sizes remains lacking. In this work, we derive an explicit upper bound for the pointwise discretization mismatch error (DME) of FNOs. Our bound characterizes the dependence of the DME on the size of the base discretization and the relative refinement between the two discretizations. In particular, for any fixed base discretization, the bound remains uniformly controlled as the finer discretization becomes arbitrarily fine. Thus, refining the finer discretization does not amplify the mismatch beyond a bound determined by the base discretization. Moreover, the bound converges to zero as the base discretization is refined. These results provide a quantitative characterization of how discretization refinement affects pointwise mismatch in FNOs. Numerical experiments on representative FNO settings support our theoretical results and demonstrate the refinement-uniform behavior predicted by the bound.