Structure-preserving model reduction of partially observed differential equations: molecular dynamics and beyond

Model reduction is a major issue for control, optimization and simulation of large-scale systems. We present a formal procedure for model reduction of perturbed linear second-order differential equations. Second-order equations appear in a variety of physical contexts, e.g., in molecular dynamics or structural mechanics to mention just a few. Common spatial decomposition methods such as Proper Orthogonal Decomposition, Principal Component Analysis or the Karhunen-Loève expansion aim at identifying a subspace of “high-energy” modes onto which the dynamics is projected (Galerkin projection). These modes, however, may not be relevant for the dynamics. Moreover these methods tacitly assume that all degrees of freedom can actually be observed or measured. An alternative procedure is known by the name of Balanced Truncation which is a method of model reduction for stable input-output systems. Unlike the aforementioned approaches Balanced Truncation accounts for incomplete observability. It consists in finding a coordinate transformation such that modes which are least sensitive to the external perturbation (controllability) also give the least output (observability) and therefore can be neglected. Accordingly, a dimension-reduced model is obtained by restricting the dynamics to the subspace of the best controllable and observable modes. A great advantage of the method is that it gives computable a priori error bounds; a drawback is that it typically fails to preserve the problem’s physical structure and suffers from lack of stability [1, 2]. Here we adopt the framework of port-Hamiltonian systems which covers the class of relevant problems and that allows for a generalization of Balanced Truncation to second-order problems, while preserving stability and the underlying Hamiltonian structure. The restriction to the controllable/observable subspace is done by imposing a holonomic constraint using techniques from singular perturbation theory for deterministic or stochastic differential equations. Given a quadratic Hamiltonian H : R ×R → R, we consider the system