A new study is bringing machine learning into one of earthquake engineering’s most practical—and surprisingly difficult—problems: how to reconstruct one type of seismic response spectrum when only another is available. Researchers Miao Hu and Yan-Gang Zhao of Beijing University of Technology, together with Haizhong Zhang of Yamagata University, have developed long short-term memory neural-network models designed to predict relationships among acceleration, velocity, and displacement response spectra. Their results suggest that artificial intelligence may offer engineers a faster and more reliable way to translate incomplete earthquake information into the data needed for structural design, seismic assessment, and dynamic analysis.
Response spectra are among the most important tools used to understand how buildings and other structures react to ground shaking. Instead of describing an earthquake only as a time history of ground acceleration, a response spectrum shows the maximum response of many idealized single-degree-of-freedom oscillators, each with a different natural period. In practical terms, it provides a map of how structures with different heights, stiffnesses, and vibration characteristics might respond to the same earthquake. Acceleration spectra are widely used in conventional seismic design, while velocity and displacement spectra are especially important for energy-dissipation systems, base-isolated buildings, long-period structures, and assessments of deformation demand.
The difficulty is that these spectra are closely connected but not interchangeable through a single universal formula. For an oscillator with natural circular frequency ω, pseudo-velocity and pseudo-acceleration can be related approximately through expressions such as PSV = ω·Sd and PSA = ω²·Sd, where Sd is the spectral displacement. However, actual acceleration, velocity, and displacement response spectra depend on the complete dynamic response of the oscillator, including phase relationships, damping, frequency content, duration, and characteristics of the earthquake record. The distinction becomes particularly important at short and long periods, where simplified mathematical conversions can become inaccurate or unstable. Soil conditions, earthquake magnitude, source-to-site distance, and regional characteristics further complicate the relationship.
In many engineering situations, only one spectrum is accessible. A historical record may have been processed to produce acceleration values but not velocity or displacement values. A design code may provide one spectrum while a specialized analysis requires another. Engineers can calculate the missing spectra by returning to the original ground-motion time series, but those records may be unavailable, difficult to process, or incomplete. Existing conversion models have attempted to solve the problem using regression equations and correction factors, yet their performance can vary substantially across earthquake scenarios. The new research addresses this limitation by treating the spectra as structured sequences rather than as isolated numerical points.
The central technology is the long short-term memory, or LSTM, neural network. LSTM networks are a specialized form of recurrent neural network developed to learn patterns in sequential data. Their internal memory cells and gating mechanisms allow them to retain useful information over a sequence while filtering out irrelevant fluctuations. This makes them suitable for response spectra, whose values change continuously across vibration periods and whose behavior at one period is related to neighboring portions of the curve. Rather than predicting every spectral ordinate independently, an LSTM can learn the broader shape of a spectrum and the way that shape is transformed when converting from acceleration to velocity or displacement, or in the reverse direction.
The researchers trained their models using a large dataset containing 16,660 horizontal seismic acceleration records. The records cover earthquakes with magnitudes from 4.0 to 9.0 and epicentral distances between 10 and 200 kilometers, encompassing a broad range of moderate to very large events and near-to-intermediate source distances. Measurements were collected at 338 stations representing four site classes. Site classification is crucial because near-surface geology can amplify, attenuate, or redistribute seismic energy across different frequency bands. By incorporating records from multiple site conditions, the study sought to expose the neural networks to the diversity that makes spectrum conversion challenging in real-world applications.
The models were not simply trained with arbitrary settings. Their hyperparameters—the choices that determine network architecture and learning behavior—were selected using Bayesian optimization. These parameters can include the number of LSTM units, learning rate, batch size, sequence configuration, and other training controls. Selecting them by trial and error can be inefficient, particularly when the model’s performance depends on complex interactions among several settings. Bayesian optimization instead builds a probabilistic picture of which combinations are promising, then strategically tests new configurations to improve the prediction objective. The approach is especially useful when each training experiment is computationally expensive and when the search space is too complicated for straightforward grid-based methods.
According to the study, the machine-learning models were systematically compared with previously published relationships connecting acceleration, velocity, and displacement spectra. The proposed models produced improved accuracy and more stable performance across the tested records. Stability is a significant advantage in seismic engineering: a method that performs well only for a narrow range of periods, magnitudes, or site conditions may give designers a false sense of precision. A more consistent model could help reduce errors when spectra are converted for unusual earthquake scenarios, including records affected by different geological settings or broad variations in source distance. The findings do not eliminate the need for engineering judgment, but they indicate that data-driven methods can capture relationships that conventional simplified regressions may overlook.
The potential applications extend beyond a single conversion task. A reliable spectrum-to-spectrum model could support rapid preliminary assessments after an earthquake, when engineers need estimates before complete waveform processing is available. It could assist in evaluating buildings equipped with viscous dampers, where velocity-dependent forces make spectral velocity particularly relevant. It may also improve analyses of base-isolated structures, long-period bridges, high-rise buildings, and other systems whose seismic behavior is governed by displacement demand rather than acceleration alone. In seismic hazard workflows, the models could provide an additional way to transform existing ground-motion products into formats compatible with different design procedures, analytical tools, or regional standards.
The study also highlights an important shift in earthquake engineering: artificial intelligence is increasingly being used not only to predict whether damage will occur, but to learn the physical relationships embedded in ground-motion data. The success of the LSTM approach rests on the availability of extensive, carefully characterized records and on the network’s ability to recognize patterns across the entire spectral curve. At the same time, machine-learning predictions remain dependent on the data used for training. Records outside the ranges represented in the dataset—such as unusual faulting mechanisms, extreme near-fault pulses, very soft soils, or regions with markedly different seismic characteristics—may require additional validation. The authors make the analyzed data and code available from the corresponding author upon reasonable request, providing a path for further testing and independent development.
By combining a large strong-motion dataset, sequence-based deep learning, and Bayesian hyperparameter selection, Hu, Zhao, and Zhang have created a new framework for bridging gaps among the three principal forms of response spectrum. The work is not a replacement for direct calculation from earthquake waveforms, nor does it suggest that one spectrum contains unlimited information about another. Instead, it offers a practical statistical tool for situations in which engineers must work with incomplete data. As earthquakes continue to challenge cities, infrastructure, and design standards worldwide, the ability to extract more useful information from existing measurements could become an increasingly important part of the seismic engineer’s toolkit.
Subject of Research: Machine-learning prediction of relationships among acceleration, velocity, and displacement response spectra for earthquake engineering.
Article Title: Construction of relationships among acceleration, velocity and displacement response spectra using machine learning
Article References: Hu, M., Zhao, YG. & Zhang, H. “Construction of relationships among acceleration, velocity and displacement response spectra using machine learning.” Bulletin of Earthquake Engineering (2026).
Image Credits: AI Generated
DOI: https://doi.org/10.1007/s10518-026-02590-x
Keywords: Acceleration response spectrum; velocity response spectrum; displacement response spectrum; neural networks; long short-term memory; Bayesian optimization; machine learning; seismic engineering.
