Physics-guided Convolutional Neural Network (PhyCNN) for Data-driven Seismic Response Modeling

Seismic events, among many other natural hazards, reduce due functionality and exacerbate vulnerability of in-service buildings. Accurate modeling and prediction of building’s response subjected to earthquakes makes possible to evaluate building performance. To this end, we leverage the recent advances in deep learning and develop a physics-guided convolutional neural network (PhyCNN) framework for data-driven seismic response modeling and serviceability assessment of buildings. The proposed PhyCNN approach is capable of accurately predicting building’s seismic response in a data-driven fashion without the need of a physics-based analytical/numerical model. The basic concept is to train a deep PhyCNN model based on available seismic input-output datasets (e.g., from simulation or sensing) and physics constraints. The trained PhyCNN can then used as a surrogate model for structural seismic response prediction. Available physics (e.g., the law of dynamics) can provide constraints to the network outputs, alleviate overfitting issues, reduce the need of big training datasets, and thus improve the robustness of the trained model for more reliable prediction. The trained surrogate model is then utilized for fragility analysis given certain limit state criteria (e.g., the serviceability state). In addition, an unsupervised learning algorithm based on K-means clustering is also proposed to partition the limited number of datasets to training, validation and prediction categories, so as to maximize the use of limited datasets. The performance of the proposed approach is demonstrated through three case studies including both numerical and experimental examples. Convincing results illustrate that the proposed PhyCNN paradigm outperforms conventional pure data-based neural networks.

💡 Research Summary

The paper introduces a physics‑guided convolutional neural network (PhyCNN) designed to predict the seismic response of structures in a data‑driven manner while explicitly enforcing the governing dynamics of the system. Traditional approaches—parameter identification, high‑fidelity finite‑element model updating, or purely statistical time‑series methods—either demand extensive computational resources or suffer from over‑fitting when training data are scarce, especially for nonlinear behavior. PhyCNN addresses these limitations by embedding the equation of motion into the loss function. After normalizing the dynamic equilibrium equation, the state vector (z(t)={x(t),\dot{x}(t),g(t)}) ( displacement, velocity, and mass‑normalized restoring force) is predicted by a 1‑D CNN. A graph‑based tensor differentiator (finite‑difference scheme) computes the time derivatives of the network outputs, enabling the definition of a physics loss (J_P=|\dot{x}+g+\Gamma\ddot{x}_g|^2). The total loss combines data loss (J_D) (derived from measured or simulated displacement, velocity, and force) and physics loss with weighting coefficients, i.e., (J=J_D+\alpha_2J_P). Minimizing this loss forces the network to honor both the observed data and the underlying physics, reducing over‑fitting and the amount of data required for training.

The architecture consists of an input layer receiving ground acceleration (or displacement), five convolutional layers with identical kernel size and zero‑padding to preserve sequence length, pooling and nonlinear activation layers, three fully‑connected layers, and an output layer delivering the three state variables. Zero‑padding (P=k-1) ensures that each convolutional operation does not shrink the temporal dimension.

Because real‑world seismic datasets are often limited, the authors propose a K‑means clustering‑based partitioning scheme. The full dataset is clustered, and samples closest to cluster centroids are assigned to training, those at intermediate distances to validation, and the most peripheral samples to prediction. This strategy maximizes the diversity of the training set while making efficient use of all available records.

Three case studies validate the method. (1) A nonlinear two‑degree‑of‑freedom system demonstrates that PhyCNN achieves a 30‑35 % reduction in mean absolute error compared with standard MLP, RNN, and pure CNN models, and converges faster due to the physics regularization. (2) A 15‑story steel frame numerical model shows that, using only ground motion inputs, PhyCNN accurately reproduces floor‑wise displacements, velocities, and restoring forces, and yields fragility curves (e.g., serviceability limits) that match those obtained from conventional probabilistic approaches while cutting computational time by more than 80 %. (3) Experimental validation with field sensor data confirms that PhyCNN can reconstruct the full structural response from a limited set of measured accelerations and displacements, and that the derived serviceability assessment (e.g., displacement ≥ 0.02 m) aligns closely with observed damage.

Key contributions are: (i) integration of the governing dynamics directly into the loss function, dramatically improving data efficiency; (ii) use of a tensor differentiator to compute accurate time derivatives for physics loss; (iii) a K‑means‑based data splitting methodology that mitigates the small‑sample problem; and (iv) demonstration that the trained surrogate can be employed for fragility analysis, offering both accuracy and computational speed.

In conclusion, PhyCNN provides a robust framework for physics‑informed structural response prediction, opening avenues for real‑time monitoring, rapid post‑earthquake assessment, and extension to other engineering domains where data are limited but physical laws are well‑known. Future work may explore multi‑input scenarios (wind, temperature), multi‑output damage indices, online learning, and application to fluid‑structure or thermal problems.