Point-wise Anomaly Detection via Fold-bifurcation ODE

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 learninghave 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. Published as a conference paper at ICLR 2026 than modeling how stresses accumulate over time. This limitation is often masked by window-level evaluation, where detections are counted correct if they fall anywhere within an anomaly window. Under stricter point-wise anomaly detection, which demands precise localization at every timestep, performance degrades substantially (Wang et al., 2024; Paparrizos et al., 2025; Wang et al., 2025) , highlighting the importance of point-wise evaluation as a more faithful and challenging criterion for real-world anomaly detection. Many real-world failures can arise from the accumulation of stress that drives a system toward a critical transition. Importantly, our formulation captures both gradual build-up and short, abrupt spikes within the same dynamical framework. We draw inspiration from fold-bifurcation dynamics, a classical theory in dynamical systems that explains how gradual external pressure can drive a system toward an abrupt transition from normal to failed states. In its canonical form, fold-bifurcation assumes a fixed control parameter r, which represents external pressure, and studies how stable and unstable equilibria appear or disappear as r changes. Put simply, the system remains stable until the equilibria collide and vanish, at which point a sudden collapse occurs. Adapting this principle, we reinterpret the control parameter as a time-varying stress signal S(t) extracted from a forecasting model. By integrating these stress signals with a fold-bifurcationinspired ODE, we calculate the risk state z(t), which captures how small stresses compound into a tipping point, i.e., a critical threshold beyond which the system abruptly shifts from normal to failed behavior. An anomaly is flagged when z(t) crosses a threshold calibrated solely on normal data. Building on this perspective, we propose FOLD (Fold-bifurcation based Anomaly Detection), a framework that reframes detection as modeling stress accumulation, unifying the detection of both sudden deviations and gradual drifts. A distinguishing feature of FOLD is that it can be instantiated directly from an already trained forecasting model without any additional detector training. We highlight the following contributions of this work: 1. To our knowledge, we introduce FOLD the first anomaly detection framework that leverages fold-bifurcation inspired dynamics for point-wise anomaly detection, requiring no anomaly labels and no additional detector training. 2. We provide a principled formulation of anomaly detection as stress-signal driven modeling, where stress signals are integrated through a fold-bifurcation ODE to capture how gradual pressures can accumulate and trigger sudden tipping-point transitions. 3. We conduct extensive experiments on 40 benchmarks against 34 state-of-the-art baselines, demonstrating that FOLD achieves superior performance in both threshold-dependent (e.g., Point-wise F1) and threshold-independent (e.g., VUS-PR) metrics. This validates the robustness and practical value of our framework under strict point-wise evaluation protocols. RELATED WORK 2.1 TIME-SERIES ANOMALY DETECTION Recent advances in time-series anomaly detection can be broadly categorized into two dominant paradigms (Paparrizos et al., 2025): prediction-based methods, which monitor forecasting or reconstruction errors (