Daniel Floryan

Daniel Floryan

When neural networks are trained from data to simulate the dynamics of physical systems, they encounter a persistent challenge: the long-time dynamics they produce are often unphysical or unstable. We analyze the origin of such instabilities when learning linear dynamical systems, focusing on the training dynamics. We make several analytical findings which empirical observations suggest extend to nonlinear dynamical systems. First, the rate of convergence of the training dynamics is uneven and depends on the distribution of energy in the data. As a special case, the dynamics in directions where the data have no energy cannot be learned. Second, in the unlearnable directions, the dynamics produced by the neural network depend on the weight initialization, and common weight initialization schemes can produce unstable dynamics. Third, injecting synthetic noise into the data during training adds damping to the training dynamics and can stabilize the learned simulator, though doing so undesirably biases the learned dynamics. For each contributor to instability, we suggest mitigative strategies. We also highlight important differences between learning discrete-time and continuous-time dynamics, and discuss extensions to nonlinear systems.

Daniel Floryan, Michael D. Graham

Dynamical models underpin our ability to understand and predict the behavior of natural systems. Whether dynamical models are developed from first-principles derivations or from observational data, they are predicated on our choice of state variables. The choice of state variables is driven by convenience and intuition, and in the data-driven case the observed variables are often chosen to be the state variables. The dimensionality of these variables (and consequently the dynamical models) can be arbitrarily large, obscuring the underlying behavior of the system. In truth, these variables are often highly redundant and the system is driven by a much smaller set of latent intrinsic variables. In this study, we combine the mathematical theory of manifolds with the representational capacity of neural networks to develop a method that learns a system's intrinsic state variables directly from time series data, and also learns predictive models for their dynamics. What distinguishes our method is its ability to reduce data to the intrinsic dimensionality of the nonlinear manifold they live on. This ability is enabled by the concepts of charts and atlases from the theory of manifolds, whereby a manifold is represented by a collection of patches that are sewn together -- a necessary representation to attain intrinsic dimensionality. We demonstrate this approach on several high-dimensional systems with low-dimensional behavior. The resulting framework provides the ability to develop dynamical models of the lowest possible dimension, capturing the essence of a system.