Dissipative Deep Neural Dynamical Systems
This work provides a theoretical framework for understanding the stability and energy properties of neural network-based dynamical systems, which is important for researchers and practitioners designing stable control systems or predictive models using deep learning.
This paper establishes sufficient conditions for dissipativity and local asymptotic stability in discrete-time dynamical systems parameterized by deep neural networks. By treating neural networks as pointwise affine maps, the authors analyze their local linear operators to evaluate dissipativity, estimate stationary points, and partition the state-space.
In this paper, we provide sufficient conditions for dissipativity and local asymptotic stability of discrete-time dynamical systems parametrized by deep neural networks. We leverage the representation of neural networks as pointwise affine maps, thus exposing their local linear operators and making them accessible to classical system analytic and design methods. This allows us to "crack open the black box" of the neural dynamical system's behavior by evaluating their dissipativity, and estimating their stationary points and state-space partitioning. We relate the norms of these local linear operators to the energy stored in the dissipative system with supply rates represented by their aggregate bias terms. Empirically, we analyze the variance in dynamical behavior and eigenvalue spectra of these local linear operators with varying weight factorizations, activation functions, bias terms, and depths.