Parameter Inference of Time Series by Delay Embeddings and Learning Differentiable Operators

TL;DR

This paper introduces ID-ODE, a method for inferring parameters of dynamical systems from time series data using delay embeddings and neural networks, applicable even with limited observations.

Contribution

The paper presents a novel approach combining delay embeddings and differentiable neural operators to infer system parameters from partial and full state observations.

Findings

Successfully infers parameters for Lorenz, Lorenz96, Lotka-Volterra, and double pendulum systems.

Demonstrates effectiveness on real-world Hall-effect Thruster data.

Works with limited and reconstructed state observations.

Abstract

We provide a method to identify system parameters of dynamical systems, called ID-ODE -- Inference by Differentiation and Observing Delay Embeddings. In this setting, we are given a dataset of trajectories from a dynamical system with system parameter labels. Our goal is to identify system parameters of new trajectories. The given trajectories may or may not encompass the full state of the system, and we may only observe a one-dimensional time series. In the latter case, we reconstruct the full state by using delay embeddings, and under sufficient conditions, Taken's Embedding Theorem assures us the reconstruction is diffeomorphic to the original. This allows our method to work on time series. Our method works by first learning the velocity operator (as given or reconstructed) with a neural network having both state and system parameters as variable inputs. Then on new trajectories we backpropagate prediction errors to the system parameter inputs giving us a gradient. We then use gradient descent to infer the correct system parameter. We demonstrate the efficacy of our approach on many numerical examples: the Lorenz system, Lorenz96, Lotka-Volterra Predator-Prey, and the Compound Double Pendulum. We also apply our algorithm on a real-world dataset: propulsion of the Hall-effect Thruster (HET).