cs.LGMar 18, 2026

Large-Scale 3D Ground-Motion Synthesis with Physics-Inspired Latent Operator Flow Matching

Authors: Yaozhong Shi, Grigorios Lavrentiadis, Konstantinos Tsalouchidis, Zachary E. Ross, David McCallen, Caifeng Zou, Kamyar Azizzadenesheli, Domniki Asimaki

Organizations: Division of Engineering and Applied Sciences, California Institute of Technology, Pasadena, CA, USA · School of Engineering and Applied Sciences, University at Buffalo, Buffalo, NY, USA · Lawrence Berkeley National Laboratory, Berkeley, CA, USA · Seismological Laboratory, California Institute of Technology, Pasadena, CA, USA · 

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Abstract

Earthquake hazard analysis and design of spatially distributed infrastructure, such as power grids and energy pipeline networks, require scenario-specific ground-motion time histories with realistic frequency content and spatiotemporal coherence. However, producing the large ensembles needed for uncertainty quantification with physics-based simulations is computationally intensive and impractical for engineering workflows. To address this challenge, we introduce Ground-Motion Flow (GMFlow), a physics-inspired latent operator flow matching framework that generates realistic, large-scale regional ground-motion time-histories conditioned on physical parameters. Validated on simulated earthquake scenarios in the San Francisco Bay Area, GMFlow generates spatially coherent ground motion across more than 9 million grid points in seconds, achieving a 10,000-fold speedup over the simulation workflow, which opens a path toward rapid and uncertainty-aware hazard assessment for distributed infrastructure. More broadly, GMFlow advances mesh-agnostic functional generative modeling and could potentially be extended to the synthesis of large-scale spatiotemporal physical fields in diverse scientific domains.

Figures & tables

Figure 1: Workflow for physics-based ground-motion simulation in the San Francisco Bay Area. (left) A 3D representation of the heterogeneous geological structure beneath the Bay Area, adopted from https://www.usgs.gov/media/ images/3d-geologic-model with permission. (center) The problem (computational) domain used for the numerical simulations. (right) An example of finite-fault event simulation.
Figure 2: GmFlow model architecture and two-stage synthesis pipeline. (top) The original surface wavefield uu is filtered and downsampled to a low-frequency representation ufu_{f} ; the reverse mapping is performed by a super-resolution neural operator. (bottom) An encoder–decoder neural operator encodes ufu_{f} to a latent code z1z_{1} , and decodes it back to ufu_{f} ; conditional flow matching transports z0 ⁣∼ ⁣N(0,I)z_{0}\!\sim\!\mathcal{N}(0,I) to z1z_{1} given physical conditions cc .
Figure 3: Evaluation of GmFlow on a representative Mw 4.4M_{w}\,4.4 point-source scenario. (A–B) Side-by-side comparison of the ground-truth velocity wavefield and a single stochastic realization generated by GmFlow at selected time steps (snapshots). Orange dash and solid lines represent Hayward fault and coastline, respectively. (C) Peak ground velocity (PGV; m/sm/s ), comparing the ground truth, one synthetic realization, the ensemble mean ( N=100N=100 ), and the predictive uncertainty. (D) Spatial maps of the Fourier amplitude spectrum (FAS) at 0.25 Hz (top) and 0.75 Hz (bottom). (E) Normalized cross-correlation (NCC) for spatiotemporal coherence analysis, showing the peak correlation coefficient (top) and the corresponding time lag (bottom). (F) Geographic map of the computational domain, with epicenter (purple star) and the virtual seismic profiles A1–A2 and B1–B2 indicated. (G–H) Time–distance plots along profiles A1–A2 and B1–B2.
Figure 4: Evaluation of GmFlow on a representative Mw 7.0M_{w}\,7.0 fintie-rupture scenario.
Figure 5: Station-level spectral evaluation across magnitude range. (A, C, E) Spatial distribution of four randomly selected validation stations (triangles) and the epicenter (star) for MwM_{w} 4.4, 6.0, and 7.0 scenarios. (B, D, F) Comparison of the horizontal (power-mean) and vertical Fourier Amplitude Spectra (FAS) for the ground truth (blue) and synthetic realizations (orange/red). The solid lines for the synthetics represent the geometric mean of N=100N=100 realizations, while the shaded regions indicate the ±1\pm 1 geometric standard deviation.
Figure 6: Statistical distribution and magnitude scaling laws of generated ground motions . (A) Pointwise distributions of log⁡10(PGV)\log_{10}(\mathrm{PGV}) for MwM_{w} 4.4, 6.0, and 7.0 comparing held-out data to the same amount of synthetic realizations generated under identical conditions. (B) Pointwise distributions of log⁡10(FAS)\log_{10}(\mathrm{FAS}) at 0.25, 0.5, 0.75, and 0.96 Hz. (C) PGV magnitude scaling from MwM_{w} 4.4–7.0 (0.1 magnitude increments), summarized by the spatial median per event over 100 realizations per bin. (D) Analogous magnitude scaling for FAS at the same frequencies; markers and error bars denote the ensemble geometric mean and standard deviation.
Figure 7: Frequency-dependent spectral residuals. (A–C) Residual statistics of horizontal Fourier amplitude spectra (FAS) as a function of frequency for MwM_{w} 4.4, 6.0, and 7.0, computed over 300 test events using ensembles of 100 synthetic realizations per event; markers show the mean and median, and error bars indicate the 16th/84th percentiles. (D–E) Spatial maps of the mean FAS residual at 0.25, 0.50, 0.75, and 0.96 Hz for MwM_{w} 4.4 and MwM_{w} 6.0 events. Black dash and solid lines represent Hayward fault and coastline, respectively.
Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1: Computational domain adopted for the SFBA and model parameters
Figure S2: (a) Map view of the free surface shear-wave velocity in the simulation domain, extracted from the United States Geological Survey velocity model, with the E-W red lines indicating the location of the velocity model cross-sections shown in (b); (b) Vertical cross-sections of the velocity model with the Hayward fault at 0 normal distance. (adopted from Pinilla-Ramos et al. 2025 with permission)
Figure S3: (a) Small-magnitude seismic source characterization as a double-couple point-source function defined by its focal mechanism and source time function; (b) Large-magnitude seismic source characterization following the Graves-Pitarka rupture model; (c) Example of a pure stochastic rupture realization for the Hayward fault plane, shown through its total slip, rise time and slip rate ; (d) slip rate of points at various depths within the fault-plane as a function of time, normalized to show a unit area of total slip, developed within a normalized rise time of 1 second.
Figure S4: Envelope of calculated Green’s functions, rupture plane dimensions and positions for Magnitude 7 and Magnitude 6 earthquake events
Figure S5: Fault-plane discretization with point-sources at 200-m grid, repositioned in the corresponding centroids of 1x1km sub-faults
ComponentsSNOAENOFlow Matching
ArchitecturerFNOrFNODiffusion U-Net
Layers46+6-
Batch Size2864
Epochs100200300
Learning Rate5×10−45\times 10^{-4}5×10−45\times 10^{-4}2×10−42\times 10^{-4}
Parameters871 M663 M128 M
Appendix
Table S1: Hyperparameters and training details of the SNO, AENO, and Flow Matching models.
Figure S6: NCC analysis using randomly selected reference stations for the representative Mw 4.4M_{w}~4.4 event.
Figure S7: Validation of GmFlow on a representative Mw 6.0M_{w}~6.0 finite rupture source scenario.
Figure S8: Zero-shot super-resolution. Ground-truth wavefield snapshots are provided at spatial resolution 256×128256\times 128 , whereas the synthetic samples are super-resolved to spatial resolution 512×256512\times 256 . (A, C, E) Ground-truth snapshots for representative Mw 4.4M_{w}\,4.4 , Mw 6M_{w}\,6 , and Mw 7M_{w}\,7 events, respectively. (B, D, F) Corresponding zero-shot super-resolved synthetic snapshots for the same events.
DatasetsModel ↓\downarrow Metric →\rightarrowPGVFAS-0.25HzFAS-0.5HzFAS-0.75HzFAS-0.96Hzresidual mean
Mw 4.4\text{Mw}~4.4CFM - 16M7.2⋅10−27.2\cdot 10^{-2}2.3⋅10−22.3\cdot 10^{-2}5.7⋅10−25.7\cdot 10^{-2}1.2⋅10−11.2\cdot 10^{-1}1.0⋅10−11.0\cdot 10^{-1}1.6⋅10−11.6\cdot 10^{-1}
 CFM - 128M5.2⋅10−25.2\cdot 10^{-2}1.6⋅10−21.6\cdot 10^{-2}3.9⋅10−23.9\cdot 10^{-2}1.0⋅10−11.0\cdot 10^{-1}8.3⋅10−28.3\cdot 10^{-2}1.1⋅10−11.1\cdot 10^{-1}
Mw 6.0\text{Mw}~6.0CFM - 16M6.5⋅10−26.5\cdot 10^{-2}1.2⋅10−21.2\cdot 10^{-2}9.9⋅10−29.9\cdot 10^{-2}2.7⋅10−12.7\cdot 10^{-1}3.2⋅10−13.2\cdot 10^{-1}4.4⋅10−14.4\cdot 10^{-1}
 CFM - 128M3.9⋅10−23.9\cdot 10^{-2}2.0⋅10−22.0\cdot 10^{-2}5.8⋅10−25.8\cdot 10^{-2}2.1⋅10−12.1\cdot 10^{-1}2.7⋅10−12.7\cdot 10^{-1}3.4⋅10−13.4\cdot 10^{-1}
Mw 7.0\text{Mw}~7.0CFM - 16M1.4⋅10−11.4\cdot 10^{-1}7.6⋅10−27.6\cdot 10^{-2}1.9⋅10−11.9\cdot 10^{-1}3.8⋅10−13.8\cdot 10^{-1}4.7⋅10−14.7\cdot 10^{-1}6.4⋅10−16.4\cdot 10^{-1}
 CFM - 128M8.5⋅10−28.5\cdot 10^{-2}2.5⋅10−22.5\cdot 10^{-2}1.3⋅10−11.3\cdot 10^{-1}3.2⋅10−13.2\cdot 10^{-1}4.3⋅10−14.3\cdot 10^{-1}5.6⋅10−15.6\cdot 10^{-1}
Appendix
Table S2: Model-size ablation for latent flow matching. Quantitative comparison of the 16M and 128M CFM models using PGV, FAS-based Wasserstein distance at multiple frequencies, and RMSE of the mean residual for Mw 4.4M_{w}\,4.4 , Mw 6.0M_{w}\,6.0 , and Mw 7.0M_{w}\,7.0 events. The lower the better.
DatasetsModel ↓\downarrow Metric →\rightarrowPGVFAS-0.25HzFAS-0.5HzFAS-0.75HzFAS-0.96Hzresidual mean
Mw 4.4\text{Mw}~4.4fc=0.5hz{\color[rgb]{0.75,0.5,0.25}\mathbf{5.7\cdot 10^{-2}}}2.1⋅10−22.1\cdot 10^{-2}{\color[rgb]{0.75,0.5,0.25}\mathbf{4.0\cdot 10^{-2}}}{\color[rgb]{0.75,0,0.25}\mathbf{9.6\cdot 10^{-2}}}{\color[rgb]{0.75,0,0.25}\mathbf{5.9\cdot 10^{-2}}}{\color[rgb]{0.75,0,0.25}\mathbf{1.0\cdot 10^{-1}}}
 fc=0.6hz6.2⋅10−26.2\cdot 10^{-2}2.4⋅10−22.4\cdot 10^{-2}4.8⋅10−24.8\cdot 10^{-2}1.1⋅10−11.1\cdot 10^{-1}{\color[rgb]{0.75,0.5,0.25}\mathbf{7.5\cdot 10^{-2}}}1.2⋅10−11.2\cdot 10^{-1}
 fc=0.75hz{\color[rgb]{0.75,0,0.25}\mathbf{5.2\cdot 10^{-2}}}{\color[rgb]{0.75,0.5,0.25}\mathbf{1.6\cdot 10^{-2}}}{\color[rgb]{0.75,0,0.25}\mathbf{3.9\cdot 10^{-2}}}1.0⋅10−11.0\cdot 10^{-1}8.3⋅10−28.3\cdot 10^{-2}{\color[rgb]{0.75,0.5,0.25}\mathbf{1.1\cdot 10^{-1}}}
 fc=1.0hz5.9⋅10−25.9\cdot 10^{-2}{\color[rgb]{0.75,0,0.25}\mathbf{1.5\cdot 10^{-2}}}4.1⋅10−24.1\cdot 10^{-2}{\color[rgb]{0.75,0.5,0.25}\mathbf{9.7\cdot 10^{-2}}}1.1⋅10−11.1\cdot 10^{-1}1.3⋅10−11.3\cdot 10^{-1}
Mw 6.0\text{Mw}~6.0fc=0.5hz{\color[rgb]{0.75,0,0.25}\mathbf{5.7\cdot 10^{-2}}}{\color[rgb]{0.75,0,0.25}\mathbf{5.7\cdot 10^{-3}}}{\color[rgb]{0.75,0.5,0.25}\mathbf{8.8\cdot 10^{-2}}}2.7⋅10−12.7\cdot 10^{-1}{\color[rgb]{0.75,0,0.25}\mathbf{1.8\cdot 10^{-1}}}3.5⋅10−13.5\cdot 10^{-1}
 fc=0.6hz7.6⋅10−27.6\cdot 10^{-2}7.8⋅10−37.8\cdot 10^{-3}1.2⋅10−11.2\cdot 10^{-1}2.8⋅10−12.8\cdot 10^{-1}2.9⋅10−12.9\cdot 10^{-1}4.1⋅10−14.1\cdot 10^{-1}
Appendix
Table S3: Cutoff-frequency ablation for the Physics-Aligned Subspace. Quantitative comparison of models using different cutoff frequencies fc∈{0.5,0.6,0.75,1.0} Hzf_{c}\in\{0.5,0.6,0.75,1.0\}\,\mathrm{Hz} across representative Mw 4.4M_{w}\,4.4 , Mw 6.0M_{w}\,6.0 , and Mw 7.0M_{w}\,7.0 events. Metrics include PGV, FAS-based Wasserstein distance at multiple frequencies, and RMSE of the mean residual. The best and second-best results in each column are highlighted in red and brown, respectively.
Figure S9: Frequency-dependent spectral residuals for different cutoff frequency fcf_{c}
Figure S10: Evaluation of spectral-calibrated GmFlow on a representative Mw 7.0M_{w}\,7.0 point-source scenario.
Figure S11: NCC analysis using randomly selected reference stations for the spectral-calibrated representative Mw 7M_{w}~7 event.
Figure S12: Statistical distribution and magnitude scaling laws of spectral-calibrated generated ground motions
Figure S13: Frequency-dependent spectral residual with calibrated synthetics.
DatasetsModel ↓\downarrow Metric →\rightarrowPGVFAS-0.25HzFAS-0.5HzFAS-0.75HzFAS-0.96Hzresidual mean
Mw 4.4\text{Mw}~4.4GmFlow -calibrated2.1⋅10−22.1\cdot 10^{-2}1.2⋅10−21.2\cdot 10^{-2}1.0⋅10−21.0\cdot 10^{-2}2.9⋅10−22.9\cdot 10^{-2}4.6⋅10−24.6\cdot 10^{-2}5.0⋅10−45.0\cdot 10^{-4}
 GmFlow5.2⋅10−25.2\cdot 10^{-2}1.6⋅10−21.6\cdot 10^{-2}3.9⋅10−23.9\cdot 10^{-2}1.0⋅10−11.0\cdot 10^{-1}8.3⋅10−28.3\cdot 10^{-2}1.1⋅10−11.1\cdot 10^{-1}
Mw 6.0\text{Mw}~6.0GmFlow -calibrated1.4⋅10−21.4\cdot 10^{-2}9.6⋅10−39.6\cdot 10^{-3}2.3⋅10−22.3\cdot 10^{-2}1.6⋅10−21.6\cdot 10^{-2}9.8⋅10−29.8\cdot 10^{-2}1.2⋅10−31.2\cdot 10^{-3}
 GmFlow3.9⋅10−23.9\cdot 10^{-2}2.0⋅10−22.0\cdot 10^{-2}5.8⋅10−25.8\cdot 10^{-2}2.1⋅10−12.1\cdot 10^{-1}2.7⋅10−12.7\cdot 10^{-1}3.4⋅10−13.4\cdot 10^{-1}
Mw 7.0\text{Mw}~7.0GmFlow -calibrated1.8⋅10−21.8\cdot 10^{-2}1.5⋅10−21.5\cdot 10^{-2}1.2⋅10−21.2\cdot 10^{-2}2.8⋅10−22.8\cdot 10^{-2}7.8⋅10−27.8\cdot 10^{-2}2.9⋅10−22.9\cdot 10^{-2}
 GmFlow8.5⋅10−28.5\cdot 10^{-2}2.5⋅10−22.5\cdot 10^{-2}1.3⋅10−11.3\cdot 10^{-1}3.2⋅10−13.2\cdot 10^{-1}4.3⋅10−14.3\cdot 10^{-1}5.6⋅10−15.6\cdot 10^{-1}
Appendix
Table S4: Effect of post-processing spectral calibration. Quantitative comparison of GmFlow and GmFlow -calibrated for representative Mw 4.4M_{w}\,4.4 , Mw 6.0M_{w}\,6.0 , and Mw 7.0M_{w}\,7.0 events using PGV, FAS-based Wasserstein distance at selected frequencies, and RMSE of the mean residual. The lower, the better.
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