Free Point-wise Anomaly Detection via Fold-bifurcation

Anomaly detection in time series is essential for applications from industrial monitoring to financial risk management. Recent methods --- including forecasting error models, representation learning, augmentation, and weak-label learning --- have achieved strong results for specific anomaly types such as sudden point or gradual collective anomalies. While many prior works report window-level metrics that may mask errors, several recent methods evaluate at the point level as well. Our goal is to use a stricter point-wise protocol to make masking effects explicit. We introduce FOLD (Point-wise Anomaly Detection via fold-bifurcation), a framework that reframes detection as tracking a system’s proximity to a critical transition. FOLD extracts stress signals from a forecasting model and integrates them with a fold-bifurcation inspired ODE to produce the risk state, flagging anomalies once it crosses a threshold calibrated on normal data. This requires no anomaly labels and no additional detector training, enabling a parameter-free and efficient detection process. By modeling anomalies as stress accumulation toward a tipping point, FOLD naturally aligns with point-wise detection, providing a unifying and interpretable perspective that complements type-specific methods. Experiments on 40 benchmarks against 34 state-of-the-art baselines show that FOLD achieves competitive or superior performance, with particular strength under strict point-wise evaluation.