Fourier Neural Operators (FNOs) have shown promise for solving partial differential equations (PDEs).They typically employ separate parameters for different frequency modes for kernel integral in Fourier space, which, yet, usually results in an undesirably large number of parameters when solving high-dimensional PDEs.An alternative is to abandon the frequency modes exceeding a predefined threshold, which, however, limits the FNOs' ability to represent high-frequency details and poses non-trivial challenges for hyper-parameter specification. To address these, we propose the AMortized Fourier Neural Operator (AM-FNO), where an amortized neural parameterization of the kernel function is deployed to accommodate arbitrarily many frequency modes using a fixed number of parameters. We introduce two implementations using Kolmogorov–Arnold Network (KAN) and Multi-Layer Perceptron (MLP), along with orthogonal functions for MLP.We extensively evaluate our proposed method on diverse datasets from various domains and observe a 25\% and 35\% average improvement compared to competing baselines.