In this work we review some of the most recent advances in the development and application of thermodynamics-informed neural networks to the simulation of physical phenomena. These are networks that impose the fulfillment of the principles of thermodynamics by construction, so the, even in the absence of any knowledge win the laws governing the problem, forecasts will be thermodynamically admissible by construction. We will show how they outperformed classical black-box techniques and how, in combination with graph neural architectures, they provide with a great flexibility for previously unseen conditions.

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Recent field and experimental studies show that mobility patterns for humans exhibit scale-free nonlocal dynamics with heavy-tailed distributions characterized by Levy flights. To study the long-range geographical spread of infectious diseases, in this paper we propose a susceptible-infectious-susceptible epidemic model with Levy flights in which the dispersal of susceptible and infectious individuals follows a heavy-tailed jump distribution. Owing to the fractional diffusion described by a spectral fractional Neumann Laplacian, the nonlocal diffusion model can be used to address the spatiotemporal dynamics driven by the nonlocal dispersal. The primary focuses are on the existence and stability of disease-free and endemic equilibria and the impact of dispersal rate and fractional power on spatial profiles of these equilibria. A variational characterization of the basic reproduction number R₀ is obtained and its dependence on the dispersal rate and fractional power is also examined. Then R₀ is utilized to investigate the effects of spatial heterogeneity on the transmission dynamics. It is shown that R₀ serves as a threshold for determining the existence and nonexistence of an epidemic equilibrium as well as the stabilities of the disease-free and endemic equilibria. In particular, for low-risk regions, both the dispersal rate and fractional power play a critical role and are capable of altering the threshold value. Numerical simulations were performed to illustrate the theoretical results. (Based on G. Zhao & S. Ruan, J. Math Pures Appl. 2023).a. An extension to viscoelastic wave equations with time delay is also discussed.