A Deep Learning Framework for Model Reduction of Dynamical Systems

A new framework is proposed for model-order reduction in which a neural network is used to learn a reduced model of a dynamical system. As technological, social, and biological models become increasingly complex, it has become necessary to find ways of reducing the number of dynamical states of a system while still maintaining model integrity. The new approach combines ideas from two existing model-order reduction methods: proper orthogonal decomposition (POD) and balanced truncation. The previously mentioned methods reduce the number of state variables by projecting the dynamics onto a linear subspace spanned by principal components of a representative data matrix. Our proposed model-reduction method has the advantage of projecting the dynamics onto a nonlinear space. The proposed method is referred to as feature decomposition," in light of the nonlinear features extracted by way of neural networks. The method is applied to both autonomous and state-space systems. In feature decomposition for state-space systems, empirical Gramians are the representative training set on which we perform the nonlinear feature learning. It is shown that under certain assumptions, feature decomposition applied to state-space systems is equivalent to balanced truncation. Finally, the method is applied to a spreading infectious disease dynamical system and the results from our reduced order model are compared to POD.