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

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.

💡 Research Summary

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This paper addresses a critical need in computational earthquake engineering: the generation of realistic, broadband synthetic ground‑motion records for the design of nuclear reactor structures, while keeping computational costs low enough for large‑scale probabilistic assessments. The authors propose a two‑stage deep‑learning framework that couples a physics‑based surrogate model—the Multiple‑Input Fourier Neural Operator (MIFNO)—with a data‑driven generative model—a conditional Denoising Diffusion Probabilistic Model (DDPM).

The MIFNO is a neural operator that approximates the Green’s function of the three‑dimensional elastodynamic wave equation. It receives as input a three‑dimensional heterogeneous elasticity field (a(\mathbf{x})) and the point‑source parameters (location (\mathbf{x}_s) and moment‑tensor orientation (\boldsymbol{\theta}_s)). Using factorized Fourier layers (F‑FNO), the network learns a mapping from these inputs to the particle‑velocity field at the free surface. Trained on 19,200 SEM3D simulations drawn from the HEMEW‑S‑3D database (30,000 total simulations), the MIFNO reproduces low‑frequency wave arrival times and overall phase with high fidelity, but exhibits a systematic mid‑frequency spectral attenuation (especially 1–5 Hz). This bias is typical for neural networks that struggle to capture fine‑scale wave phenomena.

To compensate for this limitation, the authors introduce a conditional DDPM. In the diffusion framework, a forward process adds Gaussian noise to a clean signal over (T=1000) steps, while a learned reverse process denoises the noisy sample back to the data distribution. The DDPM is built on a UNet architecture ((\epsilon_\theta) network) and is trained to predict the noise component at each timestep. Crucially, the clean conditioning signal for the DDPM is the MIFNO output; the MIFNO weights are frozen during DDPM training. This design forces the diffusion model to preserve the physically accurate low‑frequency content supplied by the operator while learning to inject realistic high‑frequency details.

Training of the DDPM uses the same HEMEW‑S‑3D dataset, now represented as 3‑component velocity time series of 6.4 s duration. A total of 614,400 training samples are generated (32 random sensors per simulation). Training runs for 100 epochs with a batch size of 256 on four NVIDIA A100 GPUs, taking roughly 13.3 hours. Inference is extremely fast: a full wavefield prediction by MIFNO takes ~50 ms, and enhancing 512 stations with the DDPM requires about 3 minutes on a single A100.

Quantitative evaluation on 3,000 held‑out test simulations employs several metrics: relative mean absolute error (rMAE), relative root‑mean‑square error (rRMSE), frequency‑bias in three bands (low 0–1 Hz, mid 1–2 Hz, high 2–5 Hz), and the standard envelope and phase Goodness‑of‑Fit (EG, PG) used in seismology. The combined MIFNO + DDPM pipeline dramatically reduces frequency bias: the low‑, mid‑, and high‑band rFFT values shift from –0.27/–0.36/–0.44 (MIFNO alone) to +0.04/+0.08/+0.11, essentially eliminating the systematic attenuation. Phase GOF improves from 6.65 ± 1.07 to 7.52 ± 1.08, indicating better alignment of waveform peaks, while envelope GOF remains essentially unchanged. Interestingly, point‑wise errors (rMAE, rRMSE) increase slightly, likely due to residual diffusion‑induced noise that does not affect spectral or phase metrics. Visual inspection of time series and spectra (Fig. 5) confirms that the DDPM restores missing mid‑frequency energy without distorting arrival times, preserving the physically meaningful structure supplied by the MIFNO.

The authors acknowledge limitations. When the MIFNO output amplitude is very low, the DDPM struggles to extract sufficient conditioning information, occasionally degrading envelope GOF. Moreover, the current study focuses on the 0–5 Hz band; practical nuclear safety analyses often require reliable content up to 30 Hz. Extending the diffusion correction to higher frequencies, possibly with multi‑scale conditioning or hierarchical diffusion steps, is identified as future work. Additional avenues include incorporating real recorded seismograms for domain adaptation, quantifying epistemic uncertainty via Bayesian diffusion, and exploring alternative neural‑operator architectures that better capture high‑frequency physics.

In conclusion, the paper demonstrates that coupling a fast, physics‑informed neural operator with a conditional diffusion model yields synthetic ground‑motion records that are both computationally cheap and spectrally richer than either component alone. While not yet ready for direct deployment in critical infrastructure design, the approach opens a promising pathway toward real‑time, high‑fidelity earthquake simulators that can support probabilistic risk assessments and performance‑based design of nuclear facilities.