Reconstructing dynamics from sparse observations with no training on target system

TL;DR

This paper introduces a hybrid transformer and reservoir computing approach that reconstructs complex nonlinear system dynamics from sparse, single observations without any prior training data from the target system, using synthetic data from known chaotic systems.

Contribution

It presents a novel machine-learning framework that reconstructs target system dynamics without training data from the system itself, relying solely on synthetic data for training.

Findings

High reconstruction accuracy with only 20% of typical data

Effective long-term prediction of nonlinear dynamics

Works across various prototypical nonlinear systems

Abstract

In applications, an anticipated situation is where the system of interest has never been encountered before and sparse observations can be made only once. Can the dynamics be faithfully reconstructed from the limited observations without any training data? This problem defies any known traditional methods of nonlinear time-series analysis as well as existing machine-learning methods that typically require extensive data from the target system for training. We address this challenge by developing a hybrid transformer and reservoir-computing machine-learning scheme. The key idea is that, for a complex and nonlinear target system, the training of the transformer can be conducted not using any data from the target system, but with essentially unlimited synthetic data from known chaotic systems. The trained transformer is then tested with the sparse data from the target system. The output of the transformer is further fed into a reservoir computer for predicting the long-term dynamics or the attractor of the target system. The power of the proposed hybrid machine-learning framework is demonstrated using a large number of prototypical nonlinear dynamical systems, with high reconstruction accuracy even when the available data is only 20% of that required to faithfully represent the dynamical behavior of the underlying system. The framework provides a paradigm of reconstructing complex and nonlinear dynamics in the extreme situation where training data does not exist and the observations are random and sparse.