A modified Constitutive Relation Error framework to learn nonlinear constitutive laws using physics-augmented Neural Networks

A novel data-driven approach to learn constitutive law with observable data is proposed in the context of structural health monitoring (SHM). Constitutive laws are represented by a physics-augmented neural network taking strain in input and free energy in output. The physics-augmented neural networks used in this method guarantee thermodynamics consistency (here the convexity of the free energy with respect to the Green-Lagrange tensor and the stress that derives from free energy), objectivity and stress-free with zero strain. The method requires only partial strain or displacement measurements within the structure and boundary conditions, without relying on strain-stress or strain- free energy pairs. The modified Constitutive Relation Error (mCRE) is used to train neural networks through an unsupervised training process. The mCRE functional offers a rich physical sense as it can be used as a prediction quality during the inference phase of the neural network, interpreting the CRE as a model error. The approach is built on previous work on the mCRE and introduces a new minimization procedure for nonlinear state laws. As typical SHM applications may require that the neural networks should be trained online, the results of the training should not depend on user-defined hyperparameters: an important emphasis is placed on the automatic and adaptive tuning of sensitive hyperparameters. Furthermore, initialization is performed using a priori knowledge regarding the constitutive law to be learned. Different test cases were evaluated to demonstrate the effectiveness of the proposed method and results show that the proposed method achieves remarkable performances regarding the quality of the learned model, noise robustness, and low sensitivity to user- defined hyperparameters, provided that the training database is rich enough with respect to loading cases.