Fourier Neural Operators for Structural Dynamics Models: Challenges, Limitations and Advantages of Using a Spectrogram Loss

Abstract

Fourier Neural Operators (FNOs) have emerged as promising surrogates for partial differential equation solvers. In this work, we extensively tested FNOs on a variety of systems with non-linear and non-stationary properties, using a wide range of forcing functions to isolate failure mechanisms. FNOs stand out in modeling linear systems, regardless of complexity, while achieving near-perfect energy preservation and accurate spectral representation for linear dynamics. However, they fail on non-linear systems, where the failure manifests as artificial energy dissipation and manipulated frequency content. This limitation persists regardless of training dataset size, and we discuss the root cause through discretization error analysis. Comparison with LSTM as the baseline shows FNOs are superior for both linear and non-linear systems, independent of the training dataset size. We develop a spectrogram-based loss function that combines time-domain Mean Squared Error (MSE) with frequency-domain magnitude and phase errors, addressing the low-frequency bias of FNOs. This frequency-aware training eliminates artificial dissipation in linear systems and enhances the energy ratios of non-linear systems. The IEA 15MW turbine model validates our findings. Despite hundreds of degrees of freedom, FNO predictions remain accurate because the turbine behaves in a predominantly linear regime. Our findings establish that system non-linearity, rather than dimensionality or complexity, determines the success of FNO. These results provide clear guidelines for practitioners and challenge assumptions about FNOs' universality.

Figures (9)

• Figure 1: The FNO architecture employed for the 2DOF systems. The input $F(t)$ is the forcing function, and the outputs are displacements ($x_1$ and $x_2$). We used a 4-layer FNO architecture with 1024 modes for each layer. The drawing shows schematically the process that is explained in Section \ref{['sec:fno_theory']}
• Figure 2: Schematic of the 2DOF mass–spring–damper system. $k_1(t)$ can be set to be a function of time (softening), $k_2$ is a linear spring, and $k_3$ is the cubic non-linearity factor.
• Figure 3: Offshore wind turbine schematic, moment directions, and output locations above the MSL.
• Figure 4: The effect of the dataset size on the low frequency configuration error. The energy ratio scale on the y-axis is between 0.96 and 1.01, while the NRMSE is below 10%. By increasing the training dataset size, the linear cases show lower error, while the softening configuration in both linear and non-linear cases encloses smaller error due to FNO low-frequency bias.
• Figure 5: The effect of the dataset size on the high frequency configuration. The energy ratio scale on the y-axis is between 0.94 and 1.02, while the NRMSE is below 20%. For the linear cases, for both error metrics, the error is low, while the increase in the training dataset size does not provide a meaningful pattern. For the non-linear cases, the increase in the training dataset size decreases the NRMSE, and the non-linear case with softening performs better due to the reduction in the system eigenfrequency.