Dual-Path Trend Mixer: A Unified Forecasting Approach

• Dual-Path Trend Mixer is a neural architecture that decomposes multivariate time-series data into long-term trends and short-term fluctuations.
• It uses parallel subnetworks to independently extract trend and detail components, enhancing model interpretability and forecasting accuracy.
• Fusion via element-wise addition and multi-task loss functions enables robust integration of statistical features for diverse applications.

A Dual-Path Trend Mixer is a class of neural or signal-processing architectures designed to explicitly separate and jointly exploit long-term trend components and short-term fluctuation (or detail/residual) components in multivariate time-series data. This separation is typically realized through parallel processing paths or branches—each path specializing in extracting and forecasting a distinct statistical property of the series. Dual-path trend mixing has become central to state-of-the-art approaches in forecasting, modeling, and interpretability for high-dimensional temporal data, including economic indicators, traffic networks, and energy systems.

1. Mathematical Formulation of Dual-Path Decomposition

The foundational step in dual-path trend mixing is the decomposition of raw time series into distinct components. Given a multivariate time series $X = [x_1, x_2, \dots, x_t]\in\mathbb R^{N\times t}$, two primary representations are constructed:

• Trend path: Original values $x_t$, typically capturing low-frequency, long-term behavioral patterns (macroeconomic trend, slow market movement, weather baseline).
• Fluctuation path: First-difference or detrended values $\Delta x_{\tau} = x_{\tau} - x_{\tau-1}$, capturing transient dynamics and high-frequency volatility.

Some architectures generalize this decomposition using local averaging, Haar wavelets, or moving-average pooling as in MDMixer (Gao et al., 13 May 2025) and DPWMixer (Qianyang et al., 30 Nov 2025). For instance, DPWMixer recursively applies Haar-QMF filters to produce a lossless pyramid of trend and detail coefficients, while PatchMixer (Gong et al., 2023) defers decomposition until a dual-head is applied to mixed convolutional features.

The decomposed signals are then processed by two specialized subnetworks to maintain statistical separation and enable dedicated modeling of each component.

2. Architectural Designs: Parallel Subnetworks and Fusion

Most dual-path architectures instantiate two parallel branches:

• Trend branch: In PFNet (Zhou et al., 2020), a multi-layer Highway-CNN (LTPM) processes $X$ to capture long-term trends; in PatchMixer, a linear forecasting head is applied to convolutionally mixed features; in MDMixer, an MLP predicts coarse-grained, temporally smoothed outputs after average pooling; in DPWMixer, a global linear map is applied on Haar approximation coefficients.
• Fluctuation/residual branch: PFNet uses Highway-CNN (SFPM) on $\Delta X$ and a small MLP for final state mixing. PatchMixer applies a nonlinear MLP forecasting head on the same base feature; DPWMixer uses a patch-based MLP-Mixer for detail coefficients; DFT (Dong et al., 9 Nov 2024) intricately applies RWKV-based temporal modeling and multi-head attention for cross-stock volatility.

Fusion is typically element-wise addition, with no extra gating: $$\hat x_{t+h} = \hat x_{t+h-1} + \Delta \hat x_{t+h}$$ (Zhou et al., 2020), or analogous summing of forecast heads (Gong et al., 2023, Qianyang et al., 30 Nov 2025).

3. Training Objectives and Multi-Task Losses

Dual-path trend mixers leverage multi-task objectives to encourage effective specialization and error consistency across both components. PFNet uses a triple loss:

• $L_{\rm main} = \|\hat x_{t+h} - x_{t+h}\|_1$ (final prediction),
• $L_{\rm trend} = \|\hat x_{t+h-1} - x_{t+h-1}\|_1$ (trend),
• $$L_{\rm fluc} = \|\Delta\hat x_{t+h} - \Delta x_{t+h}\|_1$$ (fluctuation),

with weights $(c_1, c_2)$ optimized on validation (Zhou et al., 2020).

MDMixer aligns each multi-scale output with average-pooled ground truth, combining a main MAE loss with minor alignment penalties (Gao et al., 13 May 2025). PatchMixer applies a combined MSE and MAE loss on additively fused head outputs (Gong et al., 2023). DPWMixer uses fusion weights at multiple scales determined by channel stationarity for dynamic error weighting (Qianyang et al., 30 Nov 2025).

4. Algorithmic Specializations and Efficient Implementations

Several signal-processing approaches employ dual-path trend mixing for computational efficiency and statistical robustness. For $k$th-order trend filtering, Arnold & Tibshirani (Arnold et al., 2014) describe an efficient dual-path algorithm using the difference operator $D^{(k+1)}$:

• Primal: fit piecewise-polynomial trend by minimizing squared error plus $\ell_1$ penalty,
• Dual: solve restricted least-squares via banded Cholesky or sparse QR, updating active sets in the path algorithm.

This approach enables near-linear scaling and stable factorization for large datasets, making it suitable for regularized filtering in statistical and econometric modeling.

In transformer-based settings, DST²former (Chen et al., 18 Jan 2025) organizes temporal and spatial paths as self-attention blocks, integrating with static graph compressions and cross-attention for dynamic trend synthesis in traffic networks.

5. Interpretability, Statistical Gains, and Empirical Evidence

Dual-path designs yield improved accuracy by statistical disentanglement, stronger gradient signals, and interpretability. PFNet and DFT show that explicit supervision on trend and fluctuation channels yields lower error (RSE/RAE), higher empirical correlation (CORR), and substantial improvement in portfolio/ranking metrics for stock prediction (Zhou et al., 2020, Dong et al., 9 Nov 2024).

• DFT achieves IC=0.143 vs. MASTER IC=0.055 (up to 3×–4× improvement) (Dong et al., 9 Nov 2024).
• DPWMixer demonstrates up to 11.8%–13.5% reduction in MSE due to lossless wavelet decomposition and dual-path mixing (Qianyang et al., 30 Nov 2025).
• PatchMixer yields 3.9% (MSE) and 21.2% (CNN baseline) improvements with up to 3× faster inference (Gong et al., 2023).

Ablation studies further corroborate that removing either path (trend or fluctuation) or the explicit mixing fusion increases error, e.g., +30% MSE for long-horizon forecast when time unmixing is removed in MTS-UNMixers (Zhu et al., 26 Nov 2024).

6. Interpretability and Physical Mapping in Multivariate Series

Recent architectures, notably MTS-UNMixers (Zhu et al., 26 Nov 2024), promote explicit mapping from historical to future series:

• Temporal unmixing path identifies $K$ trend/cycle bases $f_k$ and time-dependent coefficients $\alpha_k(t)$ (subject to simplex constraints).
• Channel unmixing path identifies $M$ tick-wise basis patterns $b_m$ and mixture coefficients $\beta_m(t)$.
• All bases and coefficients are shared across history and forecast, yielding physically meaningful decompositions and improved generalization.

Estimation is achieved via causal vanilla Mamba blocks in time and bi-directional Mamba blocks in channel, enforcing strict causality and efficient learning of shared representations.

7. Comparative Analysis, Limitations, and Future Directions

Dual-path trend mixing now appears as a unifying principle underlying many state-of-the-art forecasters (PFNet, DFT, DPWMixer, PatchMixer, MDMixer, DST²former, MTS-UNMixers), superseding single-path CNN/Transformer/MLP designs in accuracy, robustness, and interpretability.

Typical limitations include reliance on fixed lookback/prediction horizons, dependency on expert features or fixed filter designs, and the need for substantial memory for multi-scale or multi-head parallelization. Future enhancements may involve adaptive decomposition, multi-horizon integration, and finer-grained temporal modeling (e.g., intra-day or high-frequency regimes).

In summary, the Dual-Path Trend Mixer, by mathematically and computationally separating trend from fluctuation (or detail) signals and fusing their specially learned forecasts, constitutes an essential methodology for interpretable, high-performance time-series analysis in contemporary research (Zhou et al., 2020, Arnold et al., 2014, Chen et al., 18 Jan 2025, Dong et al., 9 Nov 2024, Qianyang et al., 30 Nov 2025, Gong et al., 2023, Gao et al., 13 May 2025, Zhu et al., 26 Nov 2024).