Physics-augmented neural networks for constitutive modeling

Structural health monitoring remains a crucial concern in engineering, given the imperatives of safety and durability. A recent practice involves establishing a dialogue between a physical structure and its numerical model to predict the health status and limit damage. This thesis focuses on the fundamental question of automatically constructing a material constitutive model, necessary for reliably predicting the structure's state. Material constitutive laws are represented by neural networks constrained to adhere to the principles of thermodynamics, through the learning of convex potentials within the framework of Generalized Standard Materials. The training of neural networks is performed in an unsupervised manner by minimizing the modified constitutive relation error (mCRE).The mCRE functional provides a rich physical sense due to the model error indicator and the ability to guarantee compliance with reliable knowledge.The method developed in this thesis is tested on various types of nonlinear behaviors (hyperelasticity, elastoplasticity, viscoplasticity). Finally, a data assimilation procedure based on Kalman filters is developed to sequentially predict the structure's state, in the case of material behavior dependent on history.