Structural health monitoring is a major concern in the field of engineering [1]. On-board sensing techniques, such as fiber optics, enable accurate in-situ measurements of mechanical strain, providing rich experimental data that can be used in data-driven approaches. Concurrently, advanced physics-based tools facilitate high-fidelity simulations of complex phenomena, contributing to a robust foundation in simulation-based engineering sciences. The development of hybrid methods that integrate both physics-based and data-based information strikes a balance between data-driven approaches (high variance, low bias, requiring extensive data) and model-based approaches (low variance, high bias, requiring minimal data) [2]. Here, an hybrid approach based on the bias-aware modified Constitutive Relation Error (mCRE) framework [3] is proposed. The main idea of this framework is to deal with the reliability of information. The reliable part of the problem (e.g. mechanical equilibrium, thermodynamics principles) is enforced in the process whereas the unreliable part (e.g. measured values or constitutive relation) is released. On the one hand, the mCRE framework is integrated into the Kalman Filter method to accurately assimilate on-the-fly measurements with a Modified Dual Kalman Filter (MDKF) algorithm [4]. On the other hand, thermodynamics-consistent neural networks are trained to minimize the associated mCRE functionnal [5] to efficiently correct model bias on the material constitutive relation in the scope of generalized standard material. The performance of the two pillars of the method will be accessed on different material behavior cases including plasticity and damage. Finally, the coupling between Kalman filtering for on-the-fly data assimilation and neural networks for model enrichment will be discussed.