Speaker
Description
Partial Differential Equations (PDEs) play a central role in modeling complex phenomena arising in diverse applications, including battery life cycles, vegetation dynamics, renewable energy systems. Alongside classical numerical discretization techniques, recent advances increasingly rely on Neural Networks, particularly Physics-Informed Neural Networks (PINNs) [1], which approximate PDEs solutions in space and time by embedding the governing equations into the training process. This talk explores the interaction between standard numerical methods and neural network frameworks.
In the first part, we present new efficient W-methods for multidimensional PDEs, based on splitting and matrix-oriented strategies [2]. Their accuracy, stability, and computational efficiency are analyzed and compared. The effectiveness of these solvers is shown through the calibration of the two-dimensional Klausmeier vegetation model using satellite data, where convolutional neural networks are trained on datasets generated by repeatedly solving the PDEs system for varying parameters values.
In the second part, we show how classical time-integration schemes can be directly embedded within neural architectures, leading to discrete-time PINNs. In the approach we propose [3], the network outputs approximate the numerical solution at successive time steps. Numerical experiments indicate that the new PINNs are competitive with existing ones and offer an efficient framework for inverse problems such as parameters estimation.
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] D. Conte, S. González-Pinto, D. Hernández-Abreu, G. Pagano. On Approximate Matrix Factorization and TASE W-methods for the time integration of parabolic Partial Differential Equations. J. Sci. Comput., 100, 34 (2024).
[3] 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).