Physics-augmented neural networks for constitutive modeling: training with the modified Constitutive Relation Error

In the context of structural health monitoring (SHM), a novel approach is presented for learning con- stitutive laws using observable data. The proposed method utilizes a data-driven approach, employing a physics-augmented neural network to represent constitutive laws. The neural network takes strain as input and outputs free energy, ensuring thermodynamic consistency by considering the convexity of the free energy with respect to the Green-Lagrange tensor and stress derived from the free energy. Notably, this approach only requires partial strain or displacement measurements within the structure and boundary conditions, eliminating the need for strain-stress or strain-free energy pairs. To train the neural networks, an unsupervised training process is employed using a modified Constitu- tive Relation Error (mCRE) as a training metric. The mCRE offers meaningful physical interpretation, serving as a prediction quality indicator during the inference phase of the neural network. The method builds upon prior work on the mCRE and introduces a new minimization procedure for nonlinear state laws. As the proposed approach is intended for online training in SHM applications, it emphasizes the elimi- nation of user-defined hyperparameters. Automatic and adaptive tuning of sensitive hyperparameters is proposed to ensure that the training results remain consistent. Additionally, the initialization process incorporates prior knowledge of the constitutive law being learned. To validate the effectiveness of the proposed method, various test cases were evaluated. The results demonstrate remarkable performance in terms of the quality of the learned model, noise robustness, and low sensitivity to user-defined hyperparameters, as long as the training database includes a suffi- cient variety of loading cases.