Constantinos Siettos
Abstract: Over the last years, machine learning has been exploited to solve both the inverse problem, i.e. that of identifying models of differential equations describing the emergent dynamics of complex dynamical systems, and the forward problem, i.e. the numerical solution of difficult problems in differential equations including stiff ODEs, DAEs and high dimensional PDEs as an alternative to classical numerical analysis methods. A bet and a challenge is to bridge theoretical and technological advances in Artificial Intelligence, numerical analysis, and mathematical physics in order to develop novel schemes that can solve both problems faster and better than it is currently done.Here, towards to this aim, aspired by the concept of physics-informed machine learning, I present how random projection neural networks can be exploited to both solve the inverse and forward problems of nonlinear DAEs and PDEs of complex nonlinear dynamics, thus outperforming in several cases deep-learning and classical numerical analysis methods. Furthemore, I show how one can bridge this framework with the arsenal of numerical bifurcation analysis toolkit and matrix-free mathods in the Krylov subspace to systematicalky study the complex emergent dynamics of large-scale systems. The performance of the proposed framework is illustrated via several benchmark problems, and its efficiency is compared with other well-established numerical analysis methods and machine learning schemes.