This talk introduces a novel class of Physics-Informed Neural Networks (PINNs) designed for the solution of Partial Differential Equations (PDEs) arising in a variety of scientific and engineering contexts. Traditional numerical methods for PDEs, such as finite differences, finite elements, and spectral techniques, require careful space discretization and time integration. In contrast, PINNs integrate the governing equations directly into a loss function that enforces the problem residual along with initial and boundary conditions [1]. Classical PINNs typically yield continuous approximations in time and space of the PDEs solutions, though guarantees for accuracy and stability are not yet well established. To address this limitation, we explore a discrete-time formulation of PINNs that leverages the robustness of classical integration schemes. Unlike standard PINNs, this approach provides approximations that are continuous in space but discrete in time, aligning more closely with traditional numerical solvers. Existing discrete-time PINNs are based on Runge-Kutta methods, and require a separate neural network for each stage of the scheme. In this talk, we propose a reformulation of discrete-time PINNs using one-step one-stage methods, enabling the entire solution to be computed using a single neural network across the discrete grid [2, 3]. Numerical experiments show that the proposed approach enhances computational efficiency and maintains solution accuracy. Finally, we discuss the extension of discrete-time PINNs for the calibration of model parameters using real-world data on vegetation dynamics in arid and semi-arid regions. Acknowledgements: This work has been supported by the PRIN PNRR 2022 projects “BAT-MEN” (P20228C2PP) and “MatForPat”. - REFERENCES [1] 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). [2] 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). [3] C. Valentino, G. Pagano, D. Conte, B. Paternoster, F. Colace. Physics Informed Neural Networks for a Lithium-ion batteries model: a case of study. Advances in Computational Science and Engineering ACSE (2024).