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Tipping Point Forecasting in Non-Stationary Dynamics on Function Spaces

Miguel Liu-Schiaffini, Clare E. Singer, Nikola Kovachki, Sze Chai Leung, Tapio Schneider, Hyunji Jane Bae, Kamyar Azizzadenesheli, Anima Anandkumar

TL;DR

The paper tackles tipping-point forecasting in non-stationary dynamics by learning memory-enabled operators on function spaces. It introduces the Recurrent Neural Operator ($\text{RNO}$), a discretization-invariant architecture that maintains a latent memory to model pre-tipping dynamics, and a conformal prediction framework that quantifies uncertainty using physics-based constraints. Empirical results across Kuramoto–Sivashinsky, Lorenz-63, cloud cover, and airfoil problems show that RNO outperforms baselines and can forecast tipping points far in advance, even with partial physics. The approach enables zero-shot generalization to new regimes and provides a principled uncertainty bound for critical transitions with practical climate and aero-dynamics relevance.

Abstract

Tipping points are abrupt, drastic, and often irreversible changes in the evolution of non-stationary and chaotic dynamical systems. For instance, increased greenhouse gas concentrations are predicted to lead to drastic decreases in low cloud cover, referred to as a climatological tipping point. In this paper, we learn the evolution of such non-stationary dynamical systems using a novel recurrent neural operator (RNO), which learns mappings between function spaces. After training RNO on only the pre-tipping dynamics, we employ it to detect future tipping points using an uncertainty-based approach. In particular, we propose a conformal prediction framework to forecast tipping points by monitoring deviations from physics constraints (such as conserved quantities and partial differential equations), enabling forecasting of these abrupt changes along with a rigorous measure of uncertainty. We illustrate our proposed methodology on non-stationary ordinary and partial differential equations, such as the Lorenz-63 and Kuramoto-Sivashinsky equations. We also apply our methods to forecast a climate tipping point in stratocumulus cloud cover and airfoil wake and stall transitions using only limited knowledge of the governing equations. For the latter, we show that our proposed method zero-shot generalizes to forecasting multiple future tipping points under varying Reynolds numbers. In our experiments, we demonstrate that even partial or approximate physics constraints can be used to accurately forecast future tipping points.

Tipping Point Forecasting in Non-Stationary Dynamics on Function Spaces

TL;DR

The paper tackles tipping-point forecasting in non-stationary dynamics by learning memory-enabled operators on function spaces. It introduces the Recurrent Neural Operator (), a discretization-invariant architecture that maintains a latent memory to model pre-tipping dynamics, and a conformal prediction framework that quantifies uncertainty using physics-based constraints. Empirical results across Kuramoto–Sivashinsky, Lorenz-63, cloud cover, and airfoil problems show that RNO outperforms baselines and can forecast tipping points far in advance, even with partial physics. The approach enables zero-shot generalization to new regimes and provides a principled uncertainty bound for critical transitions with practical climate and aero-dynamics relevance.

Abstract

Tipping points are abrupt, drastic, and often irreversible changes in the evolution of non-stationary and chaotic dynamical systems. For instance, increased greenhouse gas concentrations are predicted to lead to drastic decreases in low cloud cover, referred to as a climatological tipping point. In this paper, we learn the evolution of such non-stationary dynamical systems using a novel recurrent neural operator (RNO), which learns mappings between function spaces. After training RNO on only the pre-tipping dynamics, we employ it to detect future tipping points using an uncertainty-based approach. In particular, we propose a conformal prediction framework to forecast tipping points by monitoring deviations from physics constraints (such as conserved quantities and partial differential equations), enabling forecasting of these abrupt changes along with a rigorous measure of uncertainty. We illustrate our proposed methodology on non-stationary ordinary and partial differential equations, such as the Lorenz-63 and Kuramoto-Sivashinsky equations. We also apply our methods to forecast a climate tipping point in stratocumulus cloud cover and airfoil wake and stall transitions using only limited knowledge of the governing equations. For the latter, we show that our proposed method zero-shot generalizes to forecasting multiple future tipping points under varying Reynolds numbers. In our experiments, we demonstrate that even partial or approximate physics constraints can be used to accurately forecast future tipping points.
Paper Structure (37 sections, 1 theorem, 21 equations, 23 figures, 8 tables)

This paper contains 37 sections, 1 theorem, 21 equations, 23 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Learning in non-stationary physical systems.
  3. Tipping point forecasting.
  4. Related Work
  5. Preliminaries
  6. Recurrent neural operator (RNO)
  7. Tipping-point Forecasting
  8. Experimental Results
  9. Learning non-stationary dynamics
  10. Cloud cover equations: tipping points from partial physics
  11. Forecasting airfoil wake and stall transitions
  12. Discussion
  13. Conclusion
  14. Lorenz-63 experimental results
  15. Learning non-stationary dynamics
  16. ...and 22 more sections

Key Result

Proposition 1

For any given $\alpha\in[0,1]$, the decision rule of calling for a tipping point at level $l(\alpha)$ has a false positive rate of at most $\alpha$, and this statement holds with probability at least $1-\delta$.

Figures (23)

  • Figure 1: (a) Tipping point in cloud fraction as a function of $\text{CO}_2$ concentration, from bulk model of the atmospheric boundary layer developed in Singer2023aSinger2023b. 3d cloud cover renderings reproduced from schneider2019possible. (b)RNO accurately forecasts tipping point $64$ seconds ahead in non-stationary Lorenz-63 system. "Predicted tipping point" is the time at which our framework predicts a tipping point will occur, and "forecast time" is the time when our framework makes this prediction.
  • Figure 2: Conceptual diagram of our proposed method applied to a flow over an airfoil with a changing angle of attack. (A) The system exhibits two tipping points, the wake and stall transitions. In this example, we consider forecasting the stall transition. (B) We train a neural operator model to learn the pre-stall dynamics of the system. Note that our method does not require access to post-tipping data at training time. (C) At inference time, we autoregressively roll-out our model's dynamics predictions and compute the predictions' physics loss at each time-step. The first time the physics loss crosses a pre-defined loss threshold, we label this as the tipping point.
  • Figure 3: Illustration of RNO auto-regressive forecasting.
  • Figure 4: Diagram of the setup for the tipping point prediction method. $U$ and $L$ correspond to the upper and lower bounds given by the CDF concentration inequality, differing from the empirical CDF by $\varepsilon$. $\alpha$ is the significance level, and $l(\alpha)$ is the corresponding loss of $L$ at $\alpha$.
  • Figure 5: For the cloud cover model, using only a mass conservation constraint (as opposed to the full system), RNO is still capable of identifying the true tipping point of the system with an error of $0.03$ years, predicting $T = 2.45$ years ahead, at $\alpha = 0.07$.
  • ...and 18 more figures

Theorems & Definitions (1)

  • Proposition