# Learning Dissipative Neural Dynamical Systems

arXiv: https://arxiv.org/abs/2309.16032 · HTML: https://mlsanity.com/paper/2309.16032/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 43 references that are in the map.
- First posted on arXiv 27 Sep 2023.

## Builds on (most cited first, 17 of 17)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [Dissipative SymODEN: Encoding Hamiltonian Dynamics with Dissipation and Control into Deep Learning](https://mlsanity.com/paper/2002.08860/) (2020)
- [Physics-Informed Machine Learning for Modeling and Control of Dynamical Systems](https://mlsanity.com/paper/2306.13867/) (2023)
- [Linear identification of nonlinear systems: A lifting technique based on the Koopman operator](https://mlsanity.com/paper/1605.04457/) (2016)
- [Dissipative Deep Neural Dynamical Systems](https://mlsanity.com/paper/2011.13492/) (2020)
- [Sequential Synthesis of Distributed Controllers for Cascade Interconnected Systems](https://mlsanity.com/paper/1902.07114/) (2019)
- [Distributed Synthesis of Local Controllers for Networked Systems with Arbitrary Interconnection Topologies](https://mlsanity.com/paper/1902.10506/) (2019)
- [Learning Koopman Operator under Dissipativity Constraints](https://mlsanity.com/paper/1911.03884/) (2019)
- [Convex Incremental Dissipativity Analysis of Nonlinear Systems - Extended version](https://mlsanity.com/paper/2006.14201/) (2020)
- [PENLAB: A MATLAB solver for nonlinear semidefinite optimization](https://mlsanity.com/paper/1311.5240/) (2013)
- [Distributed Mixed Voltage Angle and Frequency Droop Control of Microgrid Interconnections with Loss of Distribution-PMU Measurements](https://mlsanity.com/paper/1810.09132/) (2018)
- [Kernel-Based Identification with Frequency Domain Side-Information](https://mlsanity.com/paper/2111.00410/) (2021)
- [Mixed Voltage Angle and Frequency Droop Control for Transient Stability of Interconnected Microgrids with Loss of PMU Measurements](https://mlsanity.com/paper/1803.02918/) (2018)
- [From Dissipativity Theory to Compositional Abstractions of Interconnected Stochastic Hybrid Systems](https://mlsanity.com/paper/1805.08815/) (2018)
- [Data-driven identification of dissipative linear models for nonlinear systems](https://mlsanity.com/paper/1907.12640/) (2019)

## Built on by (most cited first, 5 of 5)

- [ECLipsE: Efficient Compositional Lipschitz Constant Estimation for Deep Neural Networks](https://mlsanity.com/paper/2404.04375/) (2024)
- [ECLipsE-Gen-Local: Efficient Compositional Local Lipschitz Estimates for Deep Neural Networks](https://mlsanity.com/paper/2510.05261/) (2025)
- [Hybrid Energy-Based Models for Physical AI: Provably Stable Identification of Port-Hamiltonian Dynamics](https://mlsanity.com/paper/2604.00277/) (2026)
- [Demystifying Lipschitz verification: positive matrices, negative results](https://mlsanity.com/paper/2603.28113/) (2026)
- [Data-driven discrete-time deep recurrent neural network-based modeling for dissipative systems](https://mlsanity.com/paper/2609.27186/) (2026)

## Abstract (arXiv)

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.