Solving multiscale problems with neural networks: the importance of asymptotic-preservation

Data-driven approaches have begun to gain popularity throughout science, leading to a fundamental change in the scientific method as a result of the rapid advancement of Machine Learning techniques and the enormous increase in the availability of scientific data. However, the use of conventional Deep Neural Networks (DNNs) or even conventional Physics-Informed Neural Networks (PINNs) to analyse the dynamics of complex multiscale systems can lead to incorrect inferences and predictions. This is due to the presence of small scales leading to simplified or reduced models in the system that must be satisfied during the learning process. In this talk, these problems will be addressed in light of recent results obtained in the development of Asymptotic-Preserving Neural Networks (APNNs) for hyperbolic models with diffusive scaling. A series of numerical tests will demonstrate how APNNs significantly outperform traditional DNNs and PINNs at various model scales, particularly when examining cases where only sparse information is available.