This talk focuses on the efficient numerical solution of advection-reaction-diffusion partial differential equations (PDEs) arising from modeling of real-world phenomena in various fields, such as vegetation growth in arid regions, battery charging/discharging processes [1], metallic corrosion and renewable energy production, using methods based on physics informed neural networks (PINNs) [2,3]. PINNs integrate the PDEs residuals, initial and boundary conditions into the loss function, and generally produce time- and space-continuous approximations of the numerical solution. Research on PINNs is very active and focused on improving their efficiency and reliability. To improve control over the numerical solution computation process, in this talk we focus on the so-called time-discrete PINNs. They link the loss function to the stages approximations of classical Runge-Kutta (RK) methods for initial value problems [2]. With the aim of reducing the number of operations required by such time-discrete PINNs, while increasing their flexibility, we propose a new technique to construct them, based on establishing connections between the output layer and the solutions at each step provided by one-stage implicit methods [4]. Numerical tests highlight the good potential of the new so-called step-by-step time-discrete PINNs. This talk falls within the activities of: PRIN PNRR 2022 project P20228C2PP BAT-MEN (CUP: F53D23010020001), granted by the Italian Ministry of University and Research. Bibliography [1] M. Frittelli, B. Bozzini, I. Sgura. Turing patterns in a 3D morpho-chemical bulk-surface reaction diffusion system for battery modeling. MinE, 6(2), 363-393 (2024). [2] M. Raissi, P. Perdikaris, G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019). [3] F. Colace, D. Conte, G. Pagano, B. Paternoster, C. Valentino. Physics Informed Neural Networks for a Lithium-ion batteries model: a case of study. ACSE, 2(4), 354–367 (2024). [4] C. Valentino, G. Pagano, D. Conte, B. Paternoster, F. Colace, M. Casillo. Step-by-step time discrete Physics Informed Neural Networks with application to a sustainability PDE model. Math. Comput. Simul., 230, 541–558 (2025).