Learning Deep Dissipative Dynamics

TL;DR

This paper introduces a method to transform neural network-based dynamical systems into dissipative systems, ensuring stability and energy conservation, with applications demonstrating robustness in robotics and fluid dynamics.

Contribution

It analytically solves the nonlinear KYP lemma and proposes a differentiable projection to guarantee dissipativity in neural network models.

Findings

Guarantees stability and energy conservation in learned dynamics.

Demonstrates robustness against out-of-domain inputs.

Applicable to robotics and fluid dynamics systems.

Abstract

This study challenges strictly guaranteeing ``dissipativity'' of a dynamical system represented by neural networks learned from given time-series data. Dissipativity is a crucial indicator for dynamical systems that generalizes stability and input-output stability, known to be valid across various systems including robotics, biological systems, and molecular dynamics. By analytically proving the general solution to the nonlinear Kalman-Yakubovich-Popov (KYP) lemma, which is the necessary and sufficient condition for dissipativity, we propose a differentiable projection that transforms any dynamics represented by neural networks into dissipative ones and a learning method for the transformed dynamics. Utilizing the generality of dissipativity, our method strictly guarantee stability, input-output stability, and energy conservation of trained dynamical systems. Finally, we demonstrate the robustness of our method against out-of-domain input through applications to robotic arms and fluid dynamics. Code is https://github.com/kojima-r/DeepDissipativeModel