PPDL: A Real-world Industrial User Retention Ratio Forecasting Framework Integrating Physical Priors with Deep Learning

Deep forecasting models often suffer from attenuated periodic perception and entangled trend-noise representations as network depth increases. Moreover, the widely adopted channel-independent paradigm, while improving training stability, disrupts intrinsic dynamic coordination among variables, hindering the modeling of cross-variable consistency in multivariate time series. To address these issues, we propose PESD-TSF, a physics-inspired structured decomposition framework for long-term time series forecasting that jointly emphasizes interpretability and predictive accuracy. PESD-TSF introduces three key designs. First, a Multiplicative Periodic Gating mechanism incorporates continuous-time priors to dynamically modulate signal amplitudes, preserving periodic structures across deep layers. Second, a multi-scale structured encoder integrates detrended attention with hierarchical sampling to explicitly decouple long-term trends from high-frequency variations while retaining fine-grained temporal semantics. Third, to recover disrupted inter-variable dependencies, we propose Cross-Scale Collaborative Attention (CSCA) together with an RLC regularization scheme, which reconstructs global inter-variable topology in deep feature spaces and enforces physically consistent collaboration through orthogonality and consistency constraints. Extensive experiments on benchmark datasets from multiple domains demonstrate that PESD-TSF consistently achieves state-of-the-art performance, with particularly strong gains on multivariate forecasting tasks involving complex inter-variable coupling, highlighting its superior structural modeling capability and generalization.

Multivariate time series forecasting often struggles to capture long-range dependencies due to fixed lookback windows. Retrieval-augmented forecasting addresses this by retrieving historical segments from memory, but existing approaches rely on a channel-agnostic strategy that applies the same references to all variables. This neglects inter-variable heterogeneity, where different channels exhibit distinct periodicities and spectral profiles. We propose CRAFT (Channel-wise retrieval-augmented forecasting), a novel framework that performs retrieval independently for each channel. To ensure efficiency, CRAFT adopts a two-stage pipeline: a sparse relation graph constructed in the time domain prunes irrelevant candidates, and spectral similarity in the frequency domain ranks references, emphasizing dominant periodic components while suppressing noise. Experiments on seven public benchmarks demonstrate that CRAFT outperforms state-of-the-art forecasting baselines, achieving superior accuracy with practical inference efficiency.

Real-world time series forecasting systems operate under non-stationary data streams, where forecasting performance may degrade over time. Although retraining can recover the performance, it incurs non-trivial computational and operational costs. Under limited deployment resources, the key challenge is therefore not only how to retrain but also when to retrain. While existing retraining policies often rely on indirect indicators such as drift alarms or model staleness, we instead use realized forecast errors as direct deployment feedback. In this paper, we propose PILOT (Pseudo-label-Informed Learned Online Trigger), an online retraining framework that learns when to retrain from forecast-error dynamics. Since ground-truth retraining labels are unavailable, PILOT constructs a pseudo-label from future increases in forecast error and trains a lightweight scorer to predict it from observed error states. At deployment, PILOT uses only completed forecast errors and serves as a plug-in module for arbitrary forecasting backbones without architectural modification. We evaluate PILOT under standard multivariate forecasting settings across eight benchmarks with three representative backbones---DLinear, iTransformer, and TimesNet. Across all three backbones, PILOT achieves state-of-the-art average-rank performance among retraining policies while maintaining a favorable performance--efficiency trade-off.