Deep learning for simulating the evolution of condensed matter systems at the continuum scale: methods and applications

Studying the time-evolution of complex systems is key in any scientific field and a cornerstone for the understanding of condensed matter physics. To this end, continuum models have been developed since the early times of science. Despite the huge advancements in computational methods, they can still be challenging, especially when demanding high spatial and time resolutions over large scales. In recent years, neural networks have emerged as a possible alternative to speed-up or even replace traditional numerical schemes, promising fast, yet equally accurate solutions. This Review aims to recognize the state-of-the-art of these novel approaches. The literature is inspected with a dual purpose. First, the several strategies and architectures exploited so far to tackle time-dependent evolutions via neural networks are systematized. Second, the different applications and successful uses are showcased. A general distinction is drawn between data-driven approaches, relying on the availability of large datasets of solutions, and physics-informed strategies, exploiting neural networks to solve known sets of partial differential equations. Recent approaches mixing these two methods, as well as novel concepts, are also discussed. The analysis concludes with a general evaluation of the current trends and perspective developments, contrasted with the main challenges and drawbacks still limiting the use of neural network-based approaches as effective surrogates of conventional computational methods.