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

Zibo Zhao; Zhengxiong Guan; Chaoli Zhang; Linyuan Geng; Xuanbing Zhu; Zhonglong Zheng; Fan Wu

cs.LGSep 12, 2026

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

Authors: Zibo Zhao, Zhengxiong Guan, Chaoli Zhang, Linyuan Geng, Xuanbing Zhu, Zhonglong Zheng, Fan Wu

Organizations: Zhejiang Normal University · Douyin Group · Shanghai Jiao Tong University

Abstract

In multi-channel paid user acquisition, early and accurate prediction of user retention at the channel level is crucial for optimizing budget allocation. User retention curves display a pronounced temporal pattern: an initial period of high churn transitions into long-term stability. This pattern is further characterized by regular fluctuations attributable to seasonality and exhibits high serial autocorrelation. These intrinsic properties make such curves highly suitable for analysis within a time-series forecasting framework. However, forecasting user retention ratio for large-scale short-video platform faces three major challenges: significant heterogeneity across channels, pronounced global trend of decay followed by saturation, and short look-back windows. To address these challenges, we propose PPDL, a novel forecasting framework that integrates physical priors with deep learning. We first introduce a trend-residual decomposition component. The trend is modeled using the Weibull distribution, whose parameters are learned via a Multilayer Perceptron (MLP). Secondly, for the residual component, we design an auxiliary embedding module on top of a deep learning backbone to maintain the channel identity awareness. Finally, to enhance the model's sensitivity to trends, we design a Multiscale Trend-penalized loss function. The proposed approach PPDL is validated through comprehensive experiments on industrial-scale datasets, covering three applications with an average of 30+ channels each. Experimental results show that PPDL achieves improvements across different backbones and significantly outperforms existing online solutions.

Figures & tables

Fig. 1 : The retention ratio $R(t)$ denotes the fraction of cohort users remaining active in $t$ days after activation. We call this time series of retention ratios the retention curve.

Fig. 2 : Overall structure of PPDL. Auxiliary Embedding: To enhance the model’s awareness of both different channels and temporal patterns across periodic phases, we design auxiliary embedding, which primarily include channel embedding, phase embedding, and joint embedding. PPDL is a novel framework based on backbone TSMixer, which integrates three key components: Weibull-Prior Trend Extractor, Auxiliary Embedding, and Multiscale Trend-penalized Loss Function.

Fig. 3 : App1 (Large-scale): Features a massive user base with relatively uniform demographic distributions. App2 (Medium-scale): Represents a moderate-sized user base with slight distribution skews. App3 (Small-scale): Contains a smaller user bases with notable distribution shifts.

Application	PPDL (Ours)	TSMixer	Statistical model	$\Delta$ $\mathrm{MAPE}_{a}$ (Abs.)	$\Delta$ $\mathrm{MAPE}_{a}$ (Rel. %)
App1	3.6%	5.7%	4.9%	-2.1%	-36.8%
App2	5.0%	5.9%	6.4%	-0.9%	-15.3%
App3	6.6%	8.4%	7.7%	-1.8%	-21.4%
Avg.	5.07%	6.67%	6.33%	-1.60%	-24.50%

TABLE I : Performance comparison on three production datasets. Error is measured by the weighted

\mathrm{MAPE}{a}

(%). The

\Delta

\mathrm{MAPE}{a}

(Abs.) and

\Delta

\mathrm{MAPE}_{a}

(Rel. %) metrics represent improvements relative to TSMixer. Bold denotes the best performance.

Application	PPDL -Aux. Emb. -Mult. Trend. Loss	TSMixer	$\Delta$ $\mathrm{MAPE}_{a}$ (Abs.)	$\Delta$ $\mathrm{MAPE}_{a}$ (Rel. %)
App1	4.0%	5.7%	-1.7%	-29.8%
App2	5.1%	5.9%	-0.8%	-13.6%
App3	7.2%	8.4%	-1.2%	-14.3%
Avg.	5.43%	6.67%	-1.23%	-19.23%

TABLE II : Ablation study on the trend-residual decomposition module. The results indicate an average relative reduction of 19.23% in MAPE, confirming that the decomposition module effectively enhances the accuracy of forecasting for complex decay trends.

Application	PPDL -Trend -Mult. Trend. Loss	TSMixer	$\Delta$ $\mathrm{MAPE}_{a}$ (Abs.)	$\Delta$ $\mathrm{MAPE}_{a}$ (Rel. %)
App1	4.8%	5.7%	-0.9%	-15.8%
App2	5.7%	5.9%	-0.2%	-3.4%
App3	7.5%	8.4%	-0.9%	-10.7%
Avg.	6.00%	6.67%	-0.67%	-9.97%

TABLE III : Ablation study on the auxiliary embedding module. The inclusion of auxiliary embeddings leads to an average relative reduction of 9.97% in MAPE, demonstrating the effectiveness of incorporating shared information across different applications.

Application	PPDL -Mult. Trend. Loss	TSMixer	$\Delta$ $\mathrm{MAPE}_{a}$ (Abs.)	$\Delta$ $\mathrm{MAPE}_{a}$ (Rel. %)
App1	3.8%	5.7%	-1.9%	-33.3%
App2	5.0%	5.9%	-0.9%	-15.3%
App3	6.8%	8.4%	-1.6%	-19.0%
Avg.	5.20 %	6.67%	-1.47%	-22.53%

TABLE IV : Ablation study on the combined effect of trend-residual decomposition and auxiliary Embedding module. The integrated configuration achieves an average relative reduction of 22.53% in MAPE, highlighting the importance of these components for robust trend modeling.

Application	+Trend+Aux. Emb.	+Trend	+Aux. Emb.	Rel. +Trend	Rel. +Aux. Emb.
App1	3.8 %	4.0%	4.8%	-5.0%	-20.8%
App2	5.0 %	5.1%	5.7%	-2.0%	-12.3%
App3	6.8 %	7.2%	7.5%	-5.6%	-9.3%
Avg.	5.20 %	5.43%	6.00%	-4.20%	-14.13%

TABLE V : Comparison of the combined modules against each single module. +Trend+Aux. Emb. denotes the result of the combination of the trend-residual decomposition and auxiliary Embedding module, +Trend and +Aux. Emb. denote the results of the trend-only and auxiliary-only variants, respectively. Rel. +Trend and Rel. +Aux. Emb. denote the relative improvement of the combined modules over the respective single module.

Application	PPDL	PPDL -Mult. Trend. Loss	$\Delta$ $\mathrm{MAPE}_{a}$ (Abs.)	$\Delta$ $\mathrm{MAPE}_{a}$ (Rel. %)
App1	3.6%	3.8%	-0.2%	-3.4%
App2	5.0%	5.0%	-0.0%	-0.0%
App3	6.6%	6.8%	-0.2%	-3.3%
Avg.	5.07%	5.20%	-0.13%	-2.23%

TABLE VI : Ablation study on Multiscale Trend-penalized Loss. The results show an average relative reduction of 2.23% in MAPE, confirming the effectiveness of the multiscale trend-penalized mechanism.

Application	PPDL	Statistical model	$\Delta$ $\mathrm{MAPE}_{a}$ (Abs.)	$\Delta$ $\mathrm{MAPE}_{a}$ (Rel. %)
App1	3.6%	4.9%	-1.3%	-26.5%
App2	5.0%	6.4%	-1.4%	-21.9%
App3	6.6%	7.7%	-1.1%	-14.3%
Avg.	5.07%	6.33%	-1.27%	-20.90%

TABLE VII : Comparison with Online Statistical Model. The proposed approach achieves an average reduction of 20.90% in relative error. These results highlight the advantage of personalized modeling over the statistical baseline.

Panel of Figure 11 — Fig. 4 : Comparison of PPDL and the online model on four APP1 cases: (a)–(c) one channel with three activation dates; (d) another channel on a different activation date. The blue, green, and orange curves represent PPDL, online, and the ground truth. X-axis: retention day; Y-axis: retention rate. PPDL follows the trend more robustly, with less bias and fewer fluctuations.

Panel of Figure 12 — Fig. 5 : Comparison of forecasting results between PPDL and the primary TSMixer. The figure illustrates four representative cases, each corresponding to the retention rate forecast for a combination of application, primary channel, and secondary channel under specific activation dates. The blue, green, and orange curves represent PPDL, primary TSMixer, and the ground truth. X-axis: retention day; Y-axis: retention rate.

Application	TSMixer	iTr.	TFT	PPDL(T)	PPDL(I)	PPDL(F)
App1	5.7%	4.7%	5.0%	3.6%(-36.8%)	4.5%(-4.2%)	4.5%(-10.0%)
App2	5.9%	7.9%	5.0%	5.0%(-15.2%)	5.4%(-31.6%)	4.4%(-12.0%)
App3	8.4%	19.9%	15.7%	6.6%(-21.4%)	13.2%(-33.7%)	10.4%(-33.8%)
Avg.	6.67%	10.83%	8.57%	5.07%(-23.99%)	7.70%(-28.90%)	6.43%(-24.97%)

TABLE VIII : Performance comparison of different backbone architectures under the PPDL framework. The iTr. represent primary iTransformer. PPDL(T), PPDL(I), and PPDL(F) denote the PPDL framework with TSMixer, iTransformer, and TFT backbones, respectively. Numbers in parentheses indicate the relative change in

\mathrm{MAPE}_{a}

compared to the corresponding primary backbone. Bold denotes the best performance.

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Multivariate Time Series Forecasting Long-Term Time Series Forecasting Time Series Forecasting Time Series Decomposition Multiscale Time-Series Modeling

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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.

Abstract

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