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.

学习非线性动力系统的神经动力学模型，保持模型的耗散性特性是一个困难的问题。本文通过两个阶段的学习方法，首先得到一个接近系统动力学的神经动力学模型，然后通过权重和偏置的扰动问题解决模型的耗散性和贴近非线性系统轨迹的问题，确保得到一个保证耗散性且接近非线性系统的神经动力学模型。