🌀 nonlinear sciences

Koopman early warning signals for bifurcation and rate-induced tipping

This paper proposes a unified early warning framework based on Koopman operator theory and deep learning to detect both bifurcation and rate-induced tipping in stochastic, nonautonomous systems, demonstrating superior performance over classical indicators in idealized models and Atlantic Meridional Overturning Circulation simulations.

Original authors: Juan Nathaniel, Carla Roesch, Derek DeSantis, Parvathi Kooloth, Hang Fan, Valerio Lucarini, Anastasia Romanou, Pierre Gentine

Published 2026-08-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Juan Nathaniel, Carla Roesch, Derek DeSantis, Parvathi Kooloth, Hang Fan, Valerio Lucarini, Anastasia Romanou, Pierre Gentine

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Complex systems, from the global climate to the human heart, often behave in ways that seem stable until they suddenly aren't. A lake can remain clear for decades, then flip into a murky, algae-choked state overnight. A financial market can hum along smoothly before crashing. Scientists call these sudden, irreversible shifts "tipping points." For years, researchers have searched for early warning signs that could predict these disasters before they happen. The most common theory relies on a concept called "critical slowing down." Imagine a ball resting in a bowl; if you nudge it, it wobbles back to the center. As the bowl becomes shallower and the ball nears the edge of tipping, that wobble takes longer to settle. In nature, this delay shows up as increased memory in the system's data, where today's state looks more like yesterday's, and larger swings in the data's variability. This has been the standard way to look for trouble in systems that are slowly losing their stability.

However, not all disasters happen because a system is slowly losing its balance. Sometimes, a system is perfectly stable, but the world around it changes too fast for it to keep up. If the external conditions shift faster than the system can adjust, it can be thrown off its path and crash, even though it was never inherently unstable to begin with. This is known as "rate-induced tipping." The old warning signs, which look for that slow, wobbling recovery, often fail to see this coming because the system isn't slowing down; it is simply being outrun. Distinguishing between a system that is losing its grip and one that is being overwhelmed by speed is a major challenge, especially in noisy, real-world environments like the Earth's climate.

A team of researchers has developed a new mathematical tool designed to spot both types of danger using a unified approach. Instead of relying on simple measures of memory or variance, they use a framework based on the "Koopman operator," a method that treats complex, non-linear movements as if they were simpler, linear patterns evolving in a higher-dimensional space. Think of it as trying to understand a swirling river by looking at the straight lines of light that reflect off the water's surface; the lines are easier to track, and their patterns reveal the hidden turbulence beneath. By analyzing how well a system's behavior can be described by these simplified patterns, the researchers created a "residual" signal. This signal measures the gap between what the system actually does and what a simplified model predicts it should do. When a system is healthy, this gap is small. When it is approaching a tipping point, the gap grows.

The researchers tested this idea on a variety of scenarios, ranging from simple mathematical models to complex simulations of the Atlantic Meridional Overturning Circulation, a massive system of ocean currents that acts as a global heat conveyor belt. In cases where the system was slowly losing stability, their new tool worked just as well as the traditional methods, detecting the growing gap as the system neared the edge. But the real breakthrough came in the rate-induced scenarios. In these tests, the traditional warning signs were confused, often sounding the alarm for systems that were perfectly fine or missing the danger entirely. The new Koopman-based tool, however, successfully distinguished between the safe and the doomed trajectories. It did this by incorporating the speed of the changing conditions into its analysis, effectively measuring how much the system was struggling to keep up with the moving target.

To make this tool even more powerful, the team combined it with deep learning, a type of artificial intelligence that can learn the best way to look at data without being told exactly what to look for. In high-dimensional settings, where there are many variables interacting at once, these learned patterns outperformed standard, pre-set ways of analyzing the data. When applied to simulations of the ocean currents, the method didn't just flag the danger; it provided a clear, interpretable picture of what was happening. It showed that the tipping trajectories were failing to track the shifting conditions, a signature that was invisible to older methods. The tool also revealed changes in the system's internal "memory" and the speed at which it could recover from small shocks, offering a deeper physical understanding of why the collapse was happening.

The study suggests that this unified framework could be a vital addition to the toolkit for monitoring Earth's climate and other complex systems. By being able to detect both the slow loss of stability and the sudden failure to keep pace with change, it offers a more complete picture of risk. While the results were demonstrated through simulations and idealized models, the ability to extract these clear signals from noisy data points toward a future where we might better anticipate the sudden, dramatic shifts that define our changing world. The researchers emphasize that while the method is promising, it requires careful selection of the right data inputs, and in real-world applications where we cannot run "what-if" scenarios, interpreting the signals will require further refinement. Nevertheless, the work provides a significant step forward in understanding how to listen for the subtle, often silent warnings before a system tips over.

Technical Summary: Koopman Early Warning Signals for Bifurcation and Rate-Induced Tipping

Problem Statement

Complex systems often undergo abrupt transitions, or tipping points, which can be catastrophic and irreversible. Early warning signals (EWS) are critical for anticipating these events. However, existing EWS methodologies face a fundamental limitation: most rely on the concept of critical slowing down (CSD), which manifests as increasing autocorrelation and variance near bifurcation-induced tipping (B-tipping). While effective for B-tipping, where a system loses local stability, these classical indicators often fail in rate-induced tipping (R-tipping). In R-tipping, transitions occur not because stability is lost, but because external conditions change too rapidly for the system to track its evolving attractor. In such cases, the instantaneous dynamics may remain locally stable, rendering CSD-based indicators unreliable. Furthermore, in stochastic, non-autonomous systems (common in climate science), internal variability and time-varying forcing interact, making it difficult to distinguish between different tipping mechanisms using current frameworks.

Methodology

The authors propose a unified early warning framework based on Koopman operator theory, which represents finite-dimensional nonlinear dynamics through the linear evolution of infinite-dimensional observables. The core of their approach is the Residual Koopman Mode Decomposition (ResKMD).

Theoretical Framework

Residual Definition: The framework utilizes the residual obtained from ResKMD, which measures the discrepancy between the true dynamics and their finite-dimensional approximation.
- In B-tipping (Autonomous): The residual increases as the system approaches a bifurcation because slow Koopman modes accumulate near the stability boundary, causing the finite-dimensional spectral representation to deteriorate and amplifying stochastic fluctuations. Theoretically, this residual generalizes classical CSD indicators (like AR1 and variance) within a spectral operator-theoretic framework.
- In R-tipping (Control-Aware): The authors extend the framework to non-autonomous systems by augmenting the observable space with time-varying control variables ( $u_t$ ). They define a control-aware Koopman residual.
- Mechanism: In R-tipping, the system fails to track a moving attracting branch ( $\omega^*(u_t)$ ). The authors define a tracking error ( $\bar{\omega}_t = \omega_t - \omega^*(u_t)$ ) and a branch drift ( $d_t = \omega^*(u_t) - \omega^*(u_{t+1})$ ). They prove (Theorem 1) that the control-aware residual is lower-bounded by the component of the branch drift that is unresolved by the retained Koopman subspace. Thus, an increase in the residual signals a loss of tracking capability due to rapid forcing, even if local stability is preserved.
Implementation:
- Observables: The framework is tested with both prescribed dictionaries (Random Fourier Features, Time-Delay Features) and learned embeddings via deep learning (Multi-Layer Perceptron autoencoders).
- Estimation: Finite-dimensional Koopman operators are estimated from trajectory data using Extended Dynamic Mode Decomposition (EDMD).
- Metrics: The performance is evaluated using the Area Under the Receiver Operating Characteristic Curve (AUROC) to discriminate between tipping and non-tipping trajectories, as well as qualitative rolling-window analysis of spectral gaps and decay timescales.

Key Contributions

Unified Framework: The paper presents the first unified EWS framework capable of detecting both B-tipping and R-tipping in stochastic systems using a single operator-theoretic formalism (ResKMD).
Control-Aware Extension: It introduces a control-aware formulation of the Koopman residual that explicitly accounts for time-dependent forcing, addressing the specific mechanism of rate-induced tracking failure.
Deep Learning Integration: The study demonstrates that learned embeddings (via MLP autoencoders) outperform prescribed dictionaries (RFF, TDF) in separating tipping from non-tipping trajectories, particularly in high-dimensional settings.
Interpretability: Beyond detection, the framework provides interpretable spectral diagnostics, including decay timescales ( $T_e$ ), spectral gap structures, and the distribution of slow Koopman modes, offering physical insights into resilience loss.

Results

The framework was evaluated across a hierarchy of systems:

Prototypical Models:
- B-tipping (May's harvesting, Rosenzweig-MacArthur): The Koopman residual successfully increased prior to transition, consistent with CSD. Learned observables (MLP) provided better separation than RFF.
- R-tipping (Forced saddle-node, Bautin oscillator): Classical indicators (AR1, variance, spectral gap) failed to distinguish tipping from non-tipping trajectories, often showing similar trends in both. In contrast, the control-aware ResKMD showed a substantial increase specifically for tipping trajectories, correctly identifying the loss of tracking.
Empirical Observations: The method was applied to diverse real-world datasets (cyanobacteria recovery, vocal fold phonation, plant cell ATP collapse, paleoclimate transitions, and electricity blackouts). The Koopman indicators detected transitions earlier or more accurately than classical metrics, particularly in noisy or complex settings where CSD signals were inconsistent.
Atlantic Meridional Overturning Circulation (AMOC):
- Box-Model: In B-tipping scenarios, all indicators performed well. In R-tipping scenarios, classical indicators failed (falsely characterizing non-tipping runs as tipping), while control-aware ResKMD provided a clear, sharp signal near the tipping threshold.
- Coupled GCM (NASA GISS-E2.1-G): Applied to a 10-member ensemble under the SSP2-4.5 scenario, the control-aware ResKMD (using Denmark Strait sea-ice flux as the control variable) clearly separated the two members evolving toward a weak AMOC state from the eight members remaining on the strong branch. Classical AR1 and variance showed high variability and no consistent pre-transition trend. The framework also revealed that tipping members exhibited slower decay rates and a reduced spectral gap, indicating a weakening of the restoring stability.

Significance and Claims

The authors claim that this work provides a significant step toward EWS methods that remain flexible and physically interpretable in stochastic, high-dimensional, and strongly driven systems.

Unification: By treating both loss of stability (B-tipping) and loss of tracking (R-tipping) through the lens of spectral approximation error, the framework offers a common language for analyzing diverse tipping phenomena.
Robustness: The method is shown to be robust in scenarios where classical CSD indicators fail, specifically in rate-induced regimes.
Physical Insight: The use of operator-theoretic methods allows for the extraction of interpretable spectral diagnostics (e.g., spectral gap closure, decay timescales) alongside the warning signal, bridging the gap between data-driven detection and mechanistic understanding.
Limitations: The authors modestly note that the practical performance depends on the choice of observables and control variables. They acknowledge that in R-tipping, the residual growth reflects unresolved tracking dynamics in a finite-dimensional representation rather than a universal divergence, and that identifying the most dynamically relevant forcing channel remains a nontrivial modeling problem. They also highlight the challenge of deploying these methods in real-world systems where counterfactual non-tipping realizations are typically unavailable for calibration.

In summary, the paper demonstrates that Koopman-based indicators, particularly when augmented with control variables and learned embeddings, can unify the prediction and mechanistic interpretation of different tipping phenomena in complex stochastic systems.