Neural Network Learns Earthquake Response by Tuning Itself to a Building’s Natural Frequencies

Engineers simulating how a building sways during an earthquake may soon have a new computational tool at their disposal. Researchers have developed a physics-informed neural network that embeds a structure’s natural frequencies directly into its architecture, allowing it to predict seismic dynamic responses with relative displacement errors below 2 percent, without requiring any recorded response data from the structure or manual tuning of frequency parameters. The framework, called a Natural Frequency-Based Fourier Feature Physics-Informed Neural Network, or NF-FF-PINN, was validated against eleven earthquake ground motion records and several structural models ranging from simple oscillators to a shaking-table test of a twelve-story reinforced concrete frame.

The motivation stems from a well-documented weakness of conventional physics-informed neural networks. These networks embed the governing equations of a physical system into their loss function, so they can learn solutions without large labeled datasets. But fully connected architectures suffer from what researchers call spectral bias: they learn low-frequency information quickly while neglecting high-frequency components. For structural dynamics under broadband earthquake excitation, where accurate reconstruction of high-frequency content is critical, this bias is a serious handicap. The authors cite prior work showing that a traditional PINN trained on a one-dimensional Poisson equation with a multi-frequency target solution failed to solve it accurately even after more than ten million iterations.

One existing remedy is the Fourier feature PINN, which inserts a mapping layer that transforms the time input into sine and cosine components, letting the network capture high-frequency information. The catch is that the statistical spread of the mapped frequencies is a hyperparameter that must be chosen by hand, typically from trial values, and poorly chosen values can wreck the solution. The new study takes a different philosophical route: rather than treating frequency content as a tunable design variable, it uses the physical fact that a structure’s response is governed by its own natural frequencies, which are determined uniquely by the mass and stiffness matrices through the eigenvalue problem. The method sets the Fourier feature parameters equal to these natural frequencies, turning the mapping layer into what the authors describe as a physics-consistent modal embedding rather than a generic spectrum-expansion mechanism.

The theoretical justification rests on Neural Tangent Kernel theory. In the infinite-width limit, a network with Fourier features behaves like a kernel method whose spectral density is a Gaussian function controlled by the feature parameter. Because the learning rate and final recovery accuracy of each frequency component are proportional to this spectral density, the optimal parameter for learning a given frequency is that frequency itself. Setting the feature variance to the structure’s natural frequency maximizes the kernel spectrum at that frequency, ensuring the fastest learning and highest recovery accuracy for the dominant modal component. For multi-degree-of-freedom systems, the authors embed the first three natural frequencies, since structural dynamics theory indicates that lower-order modes, particularly the first three, usually dominate the overall response of high-rise buildings.

To test the approach, the team first applied it to a single-degree-of-freedom system subjected to eleven earthquake records recommended by FEMA P-695, with ground motions interpolated using cubic splines and Newmark-method solutions serving as the reference. Trained for twenty thousand iterations without any system response data, the NF-FF-PINN kept relative L2 errors in displacement consistently below 2 percent across all records. The traditional PINN, by contrast, failed outright under earthquake excitation, with relative L2 errors as high as 93 percent. The model began converging after roughly two thousand training steps, at which point the error was around 5 percent, and it stabilized within five thousand iterations.

A direct comparison with manually tuned Fourier feature PINNs highlighted why the physical embedding matters. When the hyperparameter sigma fell between 2 and 20, the tuned models produced valid solutions with errors under 5 percent, but once sigma exceeded 30 the networks completely lost their solving capability, with errors exceeding 50 percent. Using the structural natural frequency as the parameter not only guaranteed validity but also yielded slightly faster convergence. Each configuration was repeated eleven times to average out random initialization effects. The authors characterize their method as semi tuning-free, since it replaces a fragile empirical search with a deterministic quantity derived from the structure itself.

The validation then scaled up. For a four-story shear frame, including a version retrofitted with energy-dissipation dampers on every floor, the method kept L2 errors within 2 percent and achieved coefficients of determination above 0.9995, and its predictions of damper hysteresis loops closely matched theoretical curves. A three-story steel frame with 36 effective degrees of freedom, reduced by static condensation to 12 master degrees and decomposed into twelve single-degree-of-freedom equations, also showed excellent agreement with Newmark benchmarks in displacement and story drift. A two-story steel frame with bilinear hysteretic restoring forces tested nonlinear behavior: the network tracked the onset of yielding, residual displacements after vibration ceased, and the shift of the fundamental frequency to lower values in the Fourier spectrum, with displacement, velocity, and acceleration errors all within 2 percent.

The most demanding test used shaking-table experimental data from a twelve-story reinforced concrete frame tested at Tongji University. Because the numerical model’s parameters came directly from a finite-element model without any updating or calibration against measurements, discrepancies in acceleration amplitude appeared, with experimental peaks sometimes exceeding predictions. The authors are explicit that this example is a qualitative validation, intended to show that the method reproduces overall time-history characteristics and main vibration stages rather than exact point-wise agreement, and they attribute the amplitude gaps to experimental uncertainties and the absence of parameter identification.

Robustness and speed were also quantified. When five levels of Gaussian noise were injected into the training data, the proposed method maintained accuracy and stability better than the Newmark method computed on the same noisy inputs. On an Intel Core i5 CPU with an RTX 4060 GPU, training the three-story steel structure model took about 1,258 seconds, but once trained, inference of the full response time history took just 0.064 seconds, roughly 15 percent of the Newmark computation time of 0.427 seconds, a speed-up factor of 6.7. For the nonlinear two-story frame, the gap widened to about a factor of 8, with 0.068 seconds versus 0.548 seconds, because Newmark requires iterative convergence at each time step while the trained network’s inference cost depends mainly on its size rather than the number of integration steps.

The authors note limitations alongside these results. The study focuses mainly on linear structural systems, and although the bilinear hysteresis example captured limited nonlinearity, modeling structures with material or geometric nonlinearities remains future work. In inverse problems, where natural frequencies themselves may change, the fixed frequency embedding would need an adaptive adjustment scheme that does not yet exist. Within those bounds, the framework offers a continuous-time representation of structural response that sidesteps the classical trade-off between time-step size, stability, and computational cost, and the researchers suggest that transfer learning could eventually let a trained model adapt to structures with different parameters, extending its usefulness for rapid seismic loss estimation and real-time structural health monitoring.

Subject of Research: A physics-informed neural network framework that embeds structural natural frequencies into Fourier feature mappings to predict building dynamic responses under earthquake excitation.

Article Title: Structural natural frequency-embedded Fourier feature physics-informed neural networks for earthquake dynamic response prediction

Article References: Wang, X., Du, K., Li, Z., & Luo, H. (2026). Structural natural frequency-embedded Fourier feature physics-informed neural networks for earthquake dynamic response prediction. Advanced Modeling and Simulation in Engineering Sciences, 13(1), Article 6. https://doi.org/10.1186/s40323-026-00324-x

Image Credits: AI Generated

DOI: 10.1186/s40323-026-00324-x

Keywords: physics-informed neural networks, earthquake engineering, structural dynamics, natural frequencies, Fourier features, spectral bias, seismic response prediction, Neural Tangent Kernel, Newmark method, shaking-table test, machine learning, structural health monitoring