Finite element model-informed deep learning for equivalent force estimation and full-field response calculation

The measured data in structural health monitoring are discrete in space, which means that the full-field load distribution or structural responses cannot be captured. Conventional deep-learning-based data expansion methods require a huge volume of training datasets, and they tend to be overfitted because constraints are insufficient. To address these issues, a physics-informed neural network (PINN) is developed in this study by estimating an arbitrarily distributed load to an equivalent load. Subsequently, a finite element model is used to reconstruct structural responses. The proposed PINN is divided into connected trainable data model and untrainable physics-driven model. The data-driven model in the proposed PINN is a convolutional autoencoder architecture, while the physics-informed model incorporates the impulse response matrix derived from the time-invariant system. The autoencoder is fed with the monitored data, and an equivalent load is outputted. Consecutively, the impulse response function in the physics-informed model is used to calculate the responses at the sensor positions to the equivalent load. The difference between the measured and predicted responses is formulated as the loss function to train the PINN. In this way, the physics-driven model is integrated into the data model to guide its training direction in a novel self-supervised manner, thereby avoiding the overfitting effects in network training. In addition, the unmeasured and randomly-distributed load can be estimated automatically. The proposed PINN is applied to a numerical continuous beam and a laboratory-tested three-storey frame, where only a few sensors are required for response reconstruction. Results suggest that the PINN can accurately reconstruct the full-field structural responses via the estimated equivalent load and applies to complex conditions with random and unknown load distributions.