Deep Learning for Bifurcation Detection: Extending Early Warning Signals to Dynamical Systems with Coloured Noise
Bibliographic record
Abstract
Deep learning models have demonstrated remarkable success in recognising tipping points and providing early warning signals. However, there has been limited exploration of their application to dynamical systems governed by coloured noise, which characterizes many real-world systems. In this study, we show that it is possible to leverage the normal forms of three primary types of bifurcations (fold, transcritical, and Hopf) to construct a training set that enables deep learning architectures to perform effectively. Furthermore, we showed that this approach could accommodate coloured noise by replacing white noise with red noise during the training process. To evaluate the classifier trained on red noise compared to one trained on white noise, we tested their performance on mathematical models using Receiver Operating Characteristic (ROC) curves and Area Under the Curve (AUC) scores. Our findings reveal that the deep learning architecture can be effectively trained on coloured noise inputs, as evidenced by high validation accuracy and minimal sensitivity to redness (ranging from 0.83 to 0.85). However, classifiers trained on white noise also demonstrate impressive performance in identifying tipping points in coloured time series. This is further supported by high AUC scores (ranging from 0.9 to 1) for both classifiers across different coloured stochastic time series.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".