Learning the climate of dynamical systems with state-space systems

TL;DR

The work addresses learning the climate of deterministic time series by forecasting distributions rather than trajectories. It develops a rigorous framework linking state-space learning, ergodic theory, and Perron–Frobenius dynamics to establish time-uniform density-forecast stability under structural stability with mixing or attracting measures. The main contributions show that, when these dynamical conditions hold, a $C^1$-close proxy can reproduce the true system's climate with arbitrarily high accuracy, and provide Cesàro-type convergence results for ergodic/physical measures. The numerical Lorenz experiments corroborate the theory, illustrating stable density forecasts even as point forecasts diverge, with practical implications for climate-aware forecasting in deterministic systems and guidance for designing universal, $C^1$-accurate state-space learners.

Abstract

State-space systems encompass a broad class of algorithms used for modeling and forecasting time series. For such systems to be effective, two objectives must be met: (i) accurate point forecasts of the time series must be produced, and (ii) the long-term statistical behaviour of the underlying data-generating process must be replicated. The latter objective, often referred to as learning the climate, is closely related to the task of producing accurate distribution forecasts. Empirical evidence shows that distribution forecasts are far more stable than point forecasts, which are sensitive to initial conditions. In this work, we rigorously study this phenomenon for state-space systems. The main result shows that, if the underlying data-generating process is structurally stable and possesses a mixing or an attracting measure, then a sufficiently regular initial probability distribution remains close to the true future distribution at arbitrarily long time horizons when forecasted by a $C^1-$close state-space proxy. Thus, under these conditions, learning the climate of a dynamic process with a universal family of state-space systems is feasible with arbitrarily high accuracy.

Figures (2)

• Figure 1: The first and third coordinates of the samples (a) $\mu^2_{t}$, (b) $\mu^1_{t}$, and (c) $\hat{\mu}^1_t$. Upper row: distributions at $t=1000$ ($\tau = 20$), when prediction is started. Lower row: distributions at $t=4000$ ($\tau = 80$). The mixing property of the Lorenz system causes all distributions to converge to the stationary distribution.
• Figure 2: (a) Mean distance between points in $\mu^1_{t}$ and $\mu^2_{t}$ (blue) and between points in $\mu^1_{t}$ and $\hat{\mu}^1_{t}$ (orange). The error in point predictions (orange) settles at a value comparable to the distance between unrelated initial conditions (blue) after 18 Lyapunov times of prediction. (b) Squared distance in the MMD metric between $\mu^1_{t}$ and $\mu^2_{t}$ (blue) and between $\mu^1_{t}$ and $\hat{\mu}^1_{t}$ (orange). The error in distribution predictions increases slightly with the prediction horizon, but after 9 Lyapunov times of prediction, it settles at an acceptably low value, indicating that the distributions are nearly equal. The critical values for both statistical tests are calculated at level $\alpha = 0.05.$

Theorems & Definitions (44)

• Theorem 3.1
• Lemma 3.2
• Remark 3.3
• Remark 3.4
• Theorem 3.5
• Remark 3.6
• Theorem 3.7: Convergence for ergodic and physical measures.
• Remark 3.8
• Theorem 3.9: Convergence for mixing and attracting measures.
• Theorem 4.1