Preservation of dissipativity in dimensionality reduction

TL;DR

This paper develops a method to perform dimensionality reduction on dissipative systems, ensuring the reduced system retains the Lyapunov function and dissipativity, with explicit projectors and a novel approach using monotone trees.

Contribution

It introduces an explicit construction of projectors that preserve dissipativity during reduction and extends the approach beyond manifold approximations using monotone trees.

Findings

Explicit projectors for dissipative system reduction are constructed and proven unique.

A new concept of monotone trees is introduced for dissipativity-preserving projections.

The approach extends to systems beyond manifold approximations.

Abstract

Systems with predetermined Lyapunov functions play an important role in many areas of applied mathematics, physics and engineering: dynamic optimization methods (objective functions and their modifications), machine learning (loss functions), thermodynamics and kinetics (free energy and other thermodynamic potentials), adaptive control (various objective functions, stabilization quality criteria and other Lyapunov functions). Dimensionality reduction is one of the main challenges in the modern era of big data and big models. Dimensionality reduction for systems with Lyapunov functions requires it preserving dissipativity: the reduced system must also have a Lyapunov function, which is expected to be a restriction of the original Lyapunov function on the manifold of the reduced motion. An additional complexity of the problem is that the equations of motion themselves are often unknown in detail in advance and must be determined in the course of the study, while the Lyapunov function could be determined based on incomplete data. Therefore, the projection problem arises: for a given Lyapunov function, find a field of projectors such that the reduction of it any dissipative system is again a dissipative system. In this paper, we present an explicit construction of such projectors and prove their uniqueness. We have also taken the first step beyond the approximation by manifolds. This is required in many applications. For this purpose, we introduce the concept of monotone trees and find a projection of dissipative systems onto monotone trees that preserves dissipativity.