PINNs error estimates for nonlinear equations in $\mathbb{R}$-smooth Banach spaces

In 2017, M. Raissi et al. introduced the Physics-informed neural network (PINN) approximating solutions to partial differential equations (PDEs) [29, 30]. It reduces losses related to PDE and boundary/initial conditions. In recent years, the number of papers dedicated to deep learning methods for solving PDEs, including PINNs, is constantly increasing (see, for instance, [9, 20, 24, 21] for deep learning methods and [14, 16, 17, 18, 31, 32] for PINN). Consequently, a thorough exploration of the theoretical aspects associated with PINNs is of great significance. For instance, the question arises as to why PINN's training algorithm leads us to an accurate approximation. In other words, is it possible to control total error for sufficiently small residuals/training error? In [25], S. Mishra and R. Molinaro presented an error estimation answering this question and offered an operator description of the sufficient conditions for applying such a method. Yet, these conditions, while initially outlined rather generally, in practice, require obtaining the estimate itself to verify them.