The rapid simulation of diffusion-reaction models with multiphysics coupling is a significant step for industries in the transport and energy sectors towards modeling material degradation under oxidative or humid environments. These models, often characterized by non-linearity and stiffness, pose challenges for discretization techniques due to their high computational expense, especially when solved for various boundary and initial conditions. Attempts to approximate these models using parametric a priori model order reduction methods have struggled to converge for highly non-linear systems (Ramazzotti, 2016). Recent advances in hybrid machine learning offer promising alternatives by enriching mesh-free approximations of dynamical models with empirical data. Physics-Informed Neural Networks (PINNs) have demonstrated potential for accurately simulating physical systems (Raissi et al., 2019). This work explores the application of PINNs with variant architectures to simulate coupled problems of increasing nonlinearity. To address the challenges in training posed by the multi-objective and multiscale loss functions, we implement iterative training using Multiple-PINNs (Weng et al., 2022; Niaki et al., 2021). The coupling in the system of differential equations is represented through the coupling of physics loss functions across multiple neural networks, enabling the learning of a parametric representation for a family of initial conditions. While offline training demands significant computational time, the real-time post-training inference offers a substantial reduction in computation compared to traditional discretization methods. The potential of this approach for modeling material aging problems is also discussed.