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Data-driven cold starting of good reservoirs

Lyudmila Grigoryeva, Boumediene Hamzi, Felix P. Kemeth, Yannis Kevrekidis, G Manjunath, Juan-Pablo Ortega, Matthys J. Steynberg

TL;DR

Using short histories of observations from a dynamical system, a workflow for the post-training initialization of reservoir computing systems is described, based on a map called the starting map, determined by an appropriately short history of observations that maps to a unique initial condition in the reservoir space.

Abstract

Using short histories of observations from a dynamical system, a workflow for the post-training initialization of reservoir computing systems is described. This strategy is called cold-starting, and it is based on a map called the starting map, which is determined by an appropriately short history of observations that maps to a unique initial condition in the reservoir space. The time series generated by the reservoir system using that initial state can be used to run the system in autonomous mode, to produce accurate forecasts of the time series under consideration immediately. By utilizing this map, the lengthy "washouts" that are necessary to initialize reservoir systems can be eliminated, enabling the generation of forecasts using any selection of appropriately short histories of the observations.

Data-driven cold starting of good reservoirs

TL;DR

Using short histories of observations from a dynamical system, a workflow for the post-training initialization of reservoir computing systems is described, based on a map called the starting map, determined by an appropriately short history of observations that maps to a unique initial condition in the reservoir space.

Abstract

Using short histories of observations from a dynamical system, a workflow for the post-training initialization of reservoir computing systems is described. This strategy is called cold-starting, and it is based on a map called the starting map, which is determined by an appropriately short history of observations that maps to a unique initial condition in the reservoir space. The time series generated by the reservoir system using that initial state can be used to run the system in autonomous mode, to produce accurate forecasts of the time series under consideration immediately. By utilizing this map, the lengthy "washouts" that are necessary to initialize reservoir systems can be eliminated, enabling the generation of forecasts using any selection of appropriately short histories of the observations.
Paper Structure (12 sections, 2 theorems, 20 equations, 9 figures)

This paper contains 12 sections, 2 theorems, 20 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Good reservoirs and generalized synchronizations
  3. Reservoirs and generalized synchronizations
  4. Good reservoirs
  5. Good reservoirs are indeed good
  6. The starting map and cold-starting of reservoir systems
  7. The forecasting method and implementation
  8. Empirical results
  9. Robustness in learning the starting map.
  10. Conclusion
  11. Diffusion Maps
  12. Geometric Harmonics

Key Result

Proposition 2.3

Let $F: \mathbb{R} ^N\times \mathbb{R} ^d \longrightarrow \mathbb{R} ^N$ be a good reservoir for the $\omega$-observations of the dynamical system $\phi\in {\rm Diff}^1(M)$ with generalized synchronization $f:M \longrightarrow \mathbb{R}^N$. Then:

Figures (9)

  • Figure 1: Representative trajectory of the Brusselator system sampled with $\delta t=0.2$. Initial conditions are drawn uniformly such that $u_0\sim \mathcal{U}[0,2]$ and $v_0\sim \mathcal{U}[0,3]$. (a) Trajectory in phase space of the Brusselator system. (b) $u$ variable evolution of trajectory in (a).
  • Figure 2: Representative trajectory of the Lorenz system sampled with $\delta t=0.2$. Initial conditions $u_0\sim \mathcal{N}(10,1)$, $v_0\sim \mathcal{N}(1,1)$, and $w_0 \sim \mathcal{N}(0,1)$. (a) Projection of this trajectory onto the $u$-$v$ plane. (b) $u$ variable evolution of trajectory in (a).
  • Figure 3: Autonomous path-continuing of the partial observations of the Brusselator system produced by the ESN with the readout $\widehat{h}_{ridge}$ and with the initial zero state compared to the true trajectory. The shaded area marks the part of the path which is used as a history to drive the trained ESN and the black line shows the moment when the subsequent autonomous path-continuation starts.
  • Figure 4: Diffusion maps embedding of the Brusselator system time series windows of length $5$ of the training data. $\boldsymbol{v}^{(1)}$ and $\boldsymbol{v}^{(2)}$ are the two independent diffusion maps modes spanning the data manifold. The color corresponds to one warmed-up internal state variable (here, $\mathbf{x}_0$, one of the 1024 internal reservoir states).
  • Figure 5: Representative trajectory of the test data for the Brusselator system (red). The warmup period is of length 5 (gray-shaded region). Green - predictions of the ESN with the states initialized as zero vectors and warmup used. Blue - predictions of the ESN with the states initialized with geometric harmonics (GH) method.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Theorem 3.1: Cold-started forecasting methodology