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Learning Dissipative Neural Dynamical Systems

Yuezhu Xu, S. Sivaranjani

TL;DR

The paper tackles learning neural dynamical models for nonlinear systems known to be dissipative, aiming to preserve incremental dissipativity during identification. It introduces a two-stage approach: first fit an unconstrained neural ODE to trajectory data, then minimally perturb the network weights (and subsequently adjust biases) to enforce $QSR$-incremental dissipativity via a derived matrix inequality grounded in slope-restricted activations and the S-procedure. A practical weight-perturbation optimization, plus an independent bias-tuning step with new data, yields a dissipative model that closely matches ground-truth dynamics. A Duffing oscillator case study demonstrates that small weight perturbations can enforce dissipativity with preserved trajectory fidelity, validating the method's feasibility for safe, stable neural identification in control settings.

Abstract

Consider an unknown nonlinear dynamical system that is known to be dissipative. The objective of this paper is to learn a neural dynamical model that approximates this system, while preserving the dissipativity property in the model. In general, imposing dissipativity constraints during neural network training is a hard problem for which no known techniques exist. In this work, we address the problem of learning a dissipative neural dynamical system model in two stages. First, we learn an unconstrained neural dynamical model that closely approximates the system dynamics. Next, we derive sufficient conditions to perturb the weights of the neural dynamical model to ensure dissipativity, followed by perturbation of the biases to retain the fit of the model to the trajectories of the nonlinear system. We show that these two perturbation problems can be solved independently to obtain a neural dynamical model that is guaranteed to be dissipative while closely approximating the nonlinear system.

Learning Dissipative Neural Dynamical Systems

TL;DR

The paper tackles learning neural dynamical models for nonlinear systems known to be dissipative, aiming to preserve incremental dissipativity during identification. It introduces a two-stage approach: first fit an unconstrained neural ODE to trajectory data, then minimally perturb the network weights (and subsequently adjust biases) to enforce -incremental dissipativity via a derived matrix inequality grounded in slope-restricted activations and the S-procedure. A practical weight-perturbation optimization, plus an independent bias-tuning step with new data, yields a dissipative model that closely matches ground-truth dynamics. A Duffing oscillator case study demonstrates that small weight perturbations can enforce dissipativity with preserved trajectory fidelity, validating the method's feasibility for safe, stable neural identification in control settings.

Abstract

Consider an unknown nonlinear dynamical system that is known to be dissipative. The objective of this paper is to learn a neural dynamical model that approximates this system, while preserving the dissipativity property in the model. In general, imposing dissipativity constraints during neural network training is a hard problem for which no known techniques exist. In this work, we address the problem of learning a dissipative neural dynamical system model in two stages. First, we learn an unconstrained neural dynamical model that closely approximates the system dynamics. Next, we derive sufficient conditions to perturb the weights of the neural dynamical model to ensure dissipativity, followed by perturbation of the biases to retain the fit of the model to the trajectories of the nonlinear system. We show that these two perturbation problems can be solved independently to obtain a neural dynamical model that is guaranteed to be dissipative while closely approximating the nonlinear system.
Paper Structure (9 sections, 3 theorems, 21 equations, 4 figures, 1 algorithm)

This paper contains 9 sections, 3 theorems, 21 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Learning Dissipative Neural Dynamics
  4. Identifying Neural Dynamics with No Constraints
  5. Dissipativity of Neural Dynamical Systems
  6. Algorithm to Learn Dissipative Neural Dynamics
  7. Case Study
  8. Conclusion and Future Work
  9. Appendix

Key Result

Lemma 1

For the NN eqn: neural netowrk, if there exist $\lambda_i \in \mathbb{R}_{+}, i \in \mathbb{Z}_l$ and $\lambda \in \mathbb{R}_{+}$ satisfying ineq: ML, with $P_{11}$, $P_{22}$ being symmetric matrices, and $P_{12}^T = P_{21}$, then where $\Delta z^0 = \delta(z^0_1,z^0_2)$ and $\Delta z^l = \delta(z^l_1,z^l_2)$, where $(z^0_1,z^l_1)$ and $(z^0_2,z^l_2)$ are input-output pairs for the NN defined in

Figures (4)

  • Figure 1: Approach to learn dissipative neural dynamics
  • Figure 2: Trajectories of the baseline (unconstrained) neural ODE model and the ground truth for a test input.
  • Figure 3: Trajectories of baseline model and the dissipative model after weight perturbation.
  • Figure 4: Trajectories of the dissipative model after bias adjustment, baseline model, and the ground truth.

Theorems & Definitions (4)

  • Definition 1
  • Lemma 1
  • Theorem 1
  • Lemma 2: S-Procedure