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We assume that we are given a time series of data from a dynamical system andour task is to learn the flow map of the dynamical system. We present acollection of results on how to enforce constraints coming from the dynamicalsystem in order to accelerate the training of deep neural networks to representthe flow map of the system as well as increase their predictive ability. Inparticular, we provide ways to enforce constraints during training for allthree major modes of learning, namely supervised, unsupervised andreinforcement learning. In general, the dynamic constraints need to includeterms which are analogous to memory terms in model reduction formalisms. Suchmemory terms act as a restoring force which corrects the errors committed bythe learned flow map during prediction.  For supervised learning, the constraints are added to the objective function.For the case of unsupervised learning, in particular generative adversarialnetworks, the constraints are introduced by augmenting the input of thediscriminator. Finally, for the case of reinforcement learning and inparticular actor-critic methods, the constraints are added to the rewardfunction. In addition, for the reinforcement learning case, we present a novelapproach based on homotopy of the action-value function in order to stabilizeand accelerate training. We use numerical results for the Lorenz system toillustrate the various constructions.

Enforcing constraints for time series prediction in supervised, unsupervised and reinforcement learning