# Structured generative modelling of earthquake response spectra with hierarchical latent variables in hyperbolic geometry

## TL;DR

This paper introduces a new deep learning model that uses hyperbolic geometry to generate earthquake response spectra, improving accuracy and capturing complex seismic variability.

## Contribution

The novel contribution is a hierarchical variational autoencoder with latent variables in hyperbolic space for modeling earthquake spectra.

## Key findings

- The model achieves a mean coefficient of determination of 0.961 in reconstructing earthquake response spectra.
- Hyperbolic geometry improves representational efficiency by encoding hierarchical relationships in the latent space.
- The model is practically useful for stochastic ground motion simulation and early warning systems.

## Abstract

This study presents a geometry-aware generative modelling framework for earthquake response spectra, leveraging a hierarchical variational autoencoder (HVAE) with latent variables embedded in a Poincaré ball manifold. Predicting complete ground motion response spectra is crucial for seismic hazard analysis and structural performance assessment; however, conventional machine learning models often fail to capture multi-scale physical dependencies and hierarchical uncertainity inherent in earthquake records that arise from event-to-event variablity and spatial variability. The proposed architecture is trained using source and site parameters to regularise the latent space which enables the generation of physically consistent spectral amplitudes while explicitly modelling inter- and intra-event variabilities . By exploiting hyperbolic latent geometry, the HVAE encodes hierarchical relationships into the latent space with an improved representational efficiency. Trained on a curated strong-motion dataset, the model achieves high reconstruction fidelity, with a mean coefficient of determination of 0.961 across all spectral periods. Integration into stochastic ground motion simulation and early warning pipelines demonstrates its practical utility. This work bridges geometric deep learning and seismological modelling, offering a principled, domain-aligned approach to real-time seismic risk mitigation.