# Empirical Differential Gramians for Nonlinear Model Reduction

## TL;DR

This paper introduces an empirical method for nonlinear model reduction using differential Gramians, enabling balanced truncation along fixed trajectories without solving complex PDEs, demonstrated on an RL network.

## Contribution

It presents a novel empirical approach to compute differential Gramians for nonlinear systems, simplifying the model reduction process along specific trajectories.

## Key findings

- Balanced truncation is feasible without PDEs for fixed trajectories.
- The method relies on impulse and initial state responses of the variational system.
- Demonstrated successfully on a reinforcement learning network.

## Abstract

In this paper, we present an empirical balanced truncation method for nonlinear systems with linear time-invariant input vector field components. First, we define differential reachability and observability Gramians. They are matrix valued functions of the state trajectory (i.e. the initial state and input trajectory) of the original nonlinear system, and it is difficult to find them as functions of the initial state and input. The main result of this paper is to show that for a fixed state trajectory, it is possible to compute the values of these Gramians by using impulse and initial state responses of the variational system. Therefore, balanced truncation is doable along the fixed state trajectory without solving nonlinear partial differential equations, differently from conventional nonlinear balancing methods. We further develop an approximation method, which only requires trajectories of the original nonlinear systems. Our methods are demonstrated by an RL network along a trajectory.