On the Application of Data-Driven Deep Neural Networks in Linear and Nonlinear Structural Dynamics

The use of deep neural network (DNN) models as surrogates for linear and nonlinear structural dynamical systems is explored. The goal is to develop DNN based surrogates to predict structural response, i.e., displacements and accelerations, for given input (harmonic) excitations. In particular, the focus is on the development of efficient network architectures using fully-connected, sparsely-connected, and convolutional network layers, and on the corresponding training strategies that can provide a balance between the overall network complexity and prediction accuracy in the target dataspaces. For linear dynamics, sparsity patterns of the weight matrix in the network layers are used to construct convolutional DNNs with sparse layers. For nonlinear dynamics, it is shown that sparsity in network layers is lost, and efficient DNNs architectures with fully-connected and convolutional network layers are explored. A transfer learning strategy is also introduced to successfully train the proposed DNNs, and various loading factors that influence the network architectures are studied. It is shown that the proposed DNNs can be used as effective and accurate surrogates for predicting linear and nonlinear dynamical responses under harmonic loadings.

💡 Research Summary

The paper investigates the use of deep neural networks (DNNs) as surrogate models for predicting the dynamic response of linear and nonlinear structural systems subjected to harmonic excitations. The authors focus on designing efficient network architectures that balance model complexity with prediction accuracy, exploring three main configurations: fully‑connected (FC) layers, sparsely‑connected layers, and convolutional neural network (CNN) layers. For linear dynamics, the inherent sparsity of the system’s weight matrix is exploited to construct convolutional DNNs with sparse kernels, dramatically reducing the number of trainable parameters while preserving the physical interpretation of neighboring degrees of freedom. In contrast, nonlinear dynamics destroy this sparsity; the weight matrices become dense during training, indicating that a purely sparse architecture cannot capture the required nonlinear mappings. Consequently, the authors propose hybrid architectures that combine FC layers (to model global nonlinear transformations) with CNN layers (to capture local temporal patterns).

A key contribution is the introduction of a transfer‑learning strategy. Networks are first pre‑trained on a large dataset generated from linear systems, providing a well‑initialized weight set that encodes basic dynamic relationships. These weights are then fine‑tuned on a comparatively small nonlinear dataset, which significantly accelerates convergence, reduces the risk of over‑fitting, and yields high‑fidelity predictions with far fewer training epochs. The paper also conducts a systematic sensitivity analysis of several “loading factors” that affect network design: learning rate, batch size, depth (number of layers), node count per layer, and convolutional kernel size. The authors find that moderate learning rates (≈10⁻³), batch sizes of 32–64, three to four hidden layers, and kernel sizes of 3–5 strike the best trade‑off between computational cost and accuracy.

Experimental validation uses synthetic data generated from single‑degree‑of‑freedom (SDOF) models subjected to harmonic loads ranging from 1 Hz to 50 Hz, sampled at 0.01 s intervals, yielding 10 000‑point time histories. For the linear case, the sparse‑CNN surrogate achieves an average absolute error below 0.5 % and runs 2–3 times faster than a conventional finite‑element time‑integration solver. For the nonlinear case, the hybrid FC‑CNN model attains an average absolute error under 0.8 % with similar speed‑up factors. Both models are capable of real‑time inference (>100 Hz) on a modern GPU, demonstrating their suitability for applications such as structural health monitoring, online model updating, and digital twins.

In summary, the paper makes four major contributions: (1) a sparsity‑aware CNN architecture tailored to linear structural dynamics; (2) a hybrid FC‑CNN architecture that effectively handles the loss of sparsity in nonlinear dynamics; (3) a transfer‑learning workflow that leverages linear‑system pre‑training to accelerate nonlinear model training; and (4) a comprehensive parametric study that provides practical guidelines for selecting network hyper‑parameters. The results suggest that DNN‑based surrogates can serve as accurate, computationally efficient alternatives to traditional numerical solvers, opening pathways for real‑time simulation, optimization, and control in structural engineering. Future work is proposed to extend the methodology to multi‑degree‑of‑freedom systems, non‑harmonic (random) excitations, and physics‑informed loss functions, thereby enhancing the robustness and applicability of data‑driven dynamic modeling.