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.

Multiple-Input Fourier Neural Operator (MIFNO) for source-dependent 3D elastodynamics

Numerical simulations are essential tools to evaluate the solution of the wave equation in complex settings, such as three-dimensional (3D) domains with heterogeneous properties. However, their application is limited by high computational costs and existing surrogate models lack the flexibility of numerical solvers. This work introduces the Multiple-Input Fourier Neural Operator (MIFNO) to deal with structured 3D fields representing material properties as well as vectors describing the source characteristics. The MIFNO is applied to the problem of elastic wave propagation in the Earth's crust. It is trained on the HEMEW^S-3D database containing 30000 earthquake simulations in different heterogeneous domains with random source positions and orientations. Outputs are time- and space-dependent surface wavefields. The MIFNO predictions are assessed as good to excellent based on Goodness-Of-Fit (GOF) criteria. Wave arrival times and wave fronts' propagation are very accurate since 80% of the predictions have an excellent phase GOF. The fluctuations amplitudes are good for 87% of the predictions. The envelope score is hindered by the small-scale fluctuations that are challenging to capture due to the complex physical phenomena associated with high-frequency features. Nevertheless, the MIFNO can generalize to sources located outside the training domain and it shows good generalization ability to a real complex overthrust geology. When focusing on a region of interest, transfer learning improves the accuracy with limited additional costs, since GOF scores improved by more than 1 GOF unit with only 500 additional specific samples. The MIFNO is the first surrogate model offering the flexibility of an earthquake simulator with varying sources and material properties. Its good accuracy and massive speed-up offer new perspectives to replace numerical simulations in many-query problems.

Fourier Neural Operator Surrogate Model to Predict 3D Seismic Waves Propagation

With the recent rise of neural operators, scientific machine learning offers new solutions to quantify uncertainties associated with high-fidelity numerical simulations. Traditional neural networks, such as Convolutional Neural Networks (CNN) or Physics-Informed Neural Networks (PINN), are restricted to the prediction of solutions in a predefined configuration. With neural operators, one can learn the general solution of Partial Differential Equations, such as the elastic wave equation, with varying parameters. There have been very few applications of neural operators in seismology. All of them were limited to two-dimensional settings, although the importance of three-dimensional (3D) effects is well known. In this work, we apply the Fourier Neural Operator (FNO) to predict ground motion time series from a 3D geological description. We used a high-fidelity simulation code, SEM3D, to build an extensive database of ground motions generated by 30,000 different geologies. With this database, we show that the FNO can produce accurate ground motion even when the underlying geology exhibits large heterogeneities. Intensity measures at moderate and large periods are especially well reproduced. We present the first seismological application of Fourier Neural Operators in 3D. Thanks to the generalizability of our database, we believe that our model can be used to assess the influence of geological features such as sedimentary basins on ground motion, which is paramount to evaluating site effects.