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

TL;DR

The paper addresses the need for fast, flexible surrogates of 3D elastodynamics that incorporate both heterogeneous geology and variable seismic sources. It introduces the Multiple-Input Fourier Neural Operator (MIFNO), which combines a Factorized Fourier Neural Operator for the geology with a dedicated source-branch that encodes the source position and moment tensor, enabling prediction of surface velocity fields  at high fidelity. Trained on 30,000 simulations from the HEMEWS-3D database, the MIFNO achieves good to excellent phase and envelope GOFs across diverse geologies and source configurations, with substantial speed-ups (e.g., predicting 6.4 s of ground motion in ~10 ms on a GPU). The work demonstrates strong generalization to out-of-distribution sources/geologies and shows tangible gains via transfer learning for region-specific earthquakes (e.g., Le Teil), highlighting the method's potential for many-query seismic analysis and inverse problems while preserving physical interpretability through physically rooted metrics and priors.

Abstract

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.

Figures (30)

• Figure 1: Composition of the HEMEWS-3D database. Velocity models were built from the addition of randomly chosen horizontal layers and heterogeneities 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 synthetized at the surface of the domain by a grid of virtual sensors.
• Figure 2: The F-FNO is made of one uplift block $P$, a succession of $L$ factorized Fourier layers, and three projection blocks $Q_E$, $Q_N$, $Q_Z$. The details of a factorized Fourier layer are given to show the decomposition of the FFT along each dimension.
• Figure 3: The 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 $v_K$. 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.
• Figure 4: (a) Geology in the test dataset corresponding to the predictions in panels (b) and (c). The source is located at (3.9, 2.6, -6.2km) with a strike of 298.7, dip of 85.3, and rake of 15.4. (b) Velocity time series simulated (black) and predicted (dashed red line) in the three components: East-West (E), North-South (N), Vertical (Z). (c) For the same sensor as panel (b), amplitude of the Fourier coefficients of the velocity time series.
• Figure 5: East-West component of the simulated (upper row) and predicted (center row) velocity fields for the geology illustrated in Fig. \ref{['fig:timeseries_testgeol']} at five time instants. The error between simulation and prediction is given in the lower row. The white star indicates the epicenter.