Learning Dissipative Neural Dynamical Models for Unknown Nonlinear Systems

Core Concepts

The core message of this article is to develop an approach to learn neural dynamical models that closely approximate the behavior of an unknown nonlinear system while preserving the dissipativity property of the original system.

Abstract

The article addresses the problem of learning a neural dynamical system model that approximates the behavior of an unknown nonlinear system, while ensuring that the learned model is dissipative. The key points are: The authors first train an unconstrained baseline neural ODE model to approximate the dynamics of the unknown nonlinear system. They then derive sufficient conditions on the weights of the neural network to guarantee incremental dissipativity of the learned model. This involves solving an optimization problem to minimally perturb the weights of the baseline model to enforce the dissipativity constraints. Finally, they adjust the biases of the perturbed model to retain the fit to the original nonlinear system dynamics, without losing the dissipativity guarantee. The authors demonstrate their approach on a Duffing oscillator example, showing that the final dissipative neural dynamical model closely matches the ground truth system while preserving the dissipativity property, which is not naturally inherited by the unconstrained baseline model.

Stats

The system dynamics is described by the second-order Duffing oscillator equations: ẋ1(t) = x2(t) ẋ2(t) = -ax2(t) - (b + cx1^2(t))x1(t) + u(t) where a = 1, b = 1, c = 1.

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Key Insights Distilled From

Learning Dissipative Neural Dynamical Systems

by Yuezhu Xu,S.... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2309.16032.pdf

Deeper Inquiries

How can the weight perturbation step be made more computationally efficient, especially for high-dimensional neural networks

To enhance the computational efficiency of the weight perturbation step, especially for high-dimensional neural networks, several strategies can be employed. One approach is to utilize advanced optimization techniques tailored for large-scale problems, such as stochastic gradient descent with mini-batch processing. By optimizing the weight perturbation process using parallel computing or distributed computing frameworks, the computational burden can be significantly reduced. Additionally, techniques like second-order optimization methods or quasi-Newton methods can be explored to accelerate convergence and improve efficiency. Moreover, leveraging hardware accelerators like GPUs or TPUs can expedite the computation of weight perturbations for high-dimensional neural networks.

Can the dissipativity constraints be incorporated directly into the neural network training process, rather than as a post-processing step

Incorporating dissipativity constraints directly into the neural network training process, rather than as a post-processing step, can be achieved through a constrained optimization approach. By formulating the neural network training as a constrained optimization problem, where the dissipativity constraints are explicitly included in the objective function, the network can be trained to satisfy dissipativity properties during the learning process itself. This can be implemented using specialized optimization algorithms that can handle constraints efficiently, such as interior-point methods or penalty methods. By integrating dissipativity constraints into the training phase, the neural network can learn to exhibit dissipative behavior inherently, eliminating the need for a separate post-processing step.

What other control-relevant properties, beyond dissipativity, can be enforced on the learned neural dynamical models, and how would the approach need to be modified

Beyond dissipativity, several other control-relevant properties can be enforced on the learned neural dynamical models to enhance their utility in control applications. Some of these properties include stability (such as Lyapunov stability), passivity, conicity, sector-boundedness, and L2 stability. To enforce these properties, the approach would need to be modified by incorporating additional constraints or regularization terms in the neural network training process. For instance, stability can be enforced by incorporating Lyapunov function constraints, while passivity can be achieved by imposing constraints on the energy dissipation rates. By adapting the training process to include these additional properties, the learned neural dynamical models can exhibit a broader range of control-relevant behaviors, making them more suitable for various control applications.