Speaker
Description
Physics-Informed Neural Networks (PINNs) are increasingly adopted as data-driven surrogate models for partial differential equations (PDEs), but standard formulations often rely on continuous spatio-temporal approximations and may face training instabilities in time-evolution and stiff regimes. In this contribution, we present a step-by-step time-discrete PINN methodology that produces solutions discrete in time and continuous in space, by embedding classical one-stage implicit time integrators directly into the network design. The approach establishes an explicit link between network outputs and the numerical approximations generated by implicit Euler and Crank–Nicolson schemes, leading to loss functions that enforce the time-marching structure while naturally incorporating the initial condition. We also formalize the comparison with existing Runge–Kutta-based time-discrete PINNs, highlighting how the proposed construction overcomes the need for re-training at each time step and avoids high-stage RK designs. The result is a surrogate modeling framework that inherits desirable numerical properties from the embedded integrator, while retaining the flexibility of neural approximators for fast inference over the spatial domain. The methodology is illustrated on a nonlinear diffusion–reaction sustainability PDE model, used here as a representative benchmark to discuss design choices and practical implementation aspects.