Using Neural Implicit Flow To Represent Latent Dynamics Of Canonical Systems

Over the last few years the class of the so-called Neural Operators [8] have emerged as a promising tool for many fundamental tasks in scientific machine learning (SciML), as data representation [15], time-series forecasting [19] and discovering of operators from data [11] both in data-driven and Physics-informed domains [20, 13]. Neural Operators first appeared with the introduction of Deep Neural Operators (DeepONets) [11], a new class of architectures designed to extend the capabilities of neural networks in order to better perform tasks related to operator learning. A DeepONet is composed by two subnetworks, termed trunk and branch, and essentially emulates a linear expansion, in which the trunk learns a set of basis functions for a predetermined system of coordinates, while the branch discovers penalties for these functions as they relate to the forcing variables. Alternatively, it is possible to see the branch network as a hypernetwork aimed at evaluating the last layer for the trunk [15]. Since DeepONets were first proposed, many derived and alternative approaches have been developed to address the operator learning problem.