Integrating Fourier Neural Operators with Diffusion Models to improve Spectral Representation of Synthetic Earthquake Ground Motion Response

TL;DR

Addresses the need for fast, broadband synthetic ground motions for nuclear facility design in regions with limited seismic catalogs. Proposes a two-stage approach combining the Multiple-Input Fourier Neural Operator (MIFNO) as a fast elastodynamics surrogate with a conditional Denoising Diffusion Probabilistic Model (DDPM) to correct mid-frequency content. Key contributions include showing that DDPM improves spectral fidelity and GOF metrics while preserving physical wave arrivals, enabling rapid inference across site and source conditions. The work advances physics-based probabilistic seismic hazard assessment by providing a practical surrogate that reduces reliance on expensive numerical simulations, while highlighting future work on incorporating spatial correlations and offline dataset enrichment.

Abstract

Nuclear reactor buildings must be designed to withstand the dynamic load induced by strong ground motion earthquakes. For this reason, their structural behavior must be assessed in multiple realistic ground shaking scenarios (e.g., the Maximum Credible Earthquake). However, earthquake catalogs and recorded seismograms may not always be available in the region of interest. Therefore, synthetic earthquake ground motion is progressively being employed, although with some due precautions: earthquake physics is sometimes not well enough understood to be accurately reproduced with numerical tools, and the underlying epistemic uncertainties lead to prohibitive computational costs related to model calibration. In this study, we propose an AI physics-based approach to generate synthetic ground motion, based on the combination of a neural operator that approximates the elastodynamics Green's operator in arbitrary source-geology setups, enhanced by a denoising diffusion probabilistic model. The diffusion model is trained to correct the ground motion time series generated by the neural operator. Our results show that such an approach promisingly enhances the realism of the generated synthetic seismograms, with frequency biases and Goodness-Of-Fit (GOF) scores being improved by the diffusion model. This indicates that the latter is capable to mitigate the mid-frequency spectral falloff observed in the time series generated by the neural operator. Our method showcases fast and cheap inference in different site and source conditions.

Figures (6)

• Figure 1: The https://github.com/lehmannfa/MIFNO is made of a geology branch that encodes the geology with factorized Fourier (F-Fourier) layers, and a source branch that transforms the vector of source parameters $(\bm{x}_s, \bm{\theta}_s)$ into a 4D variable $v_S$ matching the dimensions of the geology branch output $v_K$. The outputs of each branch are concatenated after elementary mathematical operations and the remaining factorized Fourier layers are applied. Uplift $P$ and projection $Q_E$, $Q_N$, $Q_Z$ blocks are the same as in the F-FNO: they embed the geology field and project the latent space into the solution space, respectively. Reprinted from Lehmann_Gatti_Clouteau_2025.
• Figure 2: Composition of the https://entrepot.recherche.data.gouv.fr/dataset.xhtml?persistentId=doi:10.57745/LAI6YU database. Velocity models were built from the addition of randomly chosen horizontal layers and heterogeneity drawn from random fields. Combined with the source position and the source orientation, they form the inputs of the neural operator. Outputs of the Spectral Element Code SEM3D are velocity wavefields synthesized at the surface of the domain by a grid of virtual sensors. Reprinted from lehmann_synthetic_2024.
• Figure 3: Diffusion process diagram. The forward process is associated to a transition kernel $q_{\phi}(\bm{v}_{\tau}\vert\bm{v}_{\tau-1})$, whereas the backward diffusion process is associated to the transition kernel $p_{\theta}(\bm{v}_{\tau}\vert\bm{v}_{\tau+1})$.
• Figure 4: Forward and backward diffusion process conditioned by the https://github.com/lehmannfa/MIFNO output (red). The https://github.com/sem3d/sem numerical solution, representing the target, is shown in black, while the DDPM output is shown in blue.
• Figure 5: Comparison between the reference numerical simulation (black dashed line), the https://github.com/lehmannfa/MIFNO prediction acting as conditioning (red line), and the signal enhanced by DDPM (blue line). The top panels (a, b, c) show the 3-component synthetic seismograms, while the bottom panels (d, e, f) show the corresponding spectra.