HydroFusion-LMF: Unified Runoff Forecasting

• HydroFusion-LMF is a unified framework for accurate decade-scale daily runoff forecasting, combining learnable trend-seasonal-residual decomposition with a heterogeneous expert ensemble.
• It employs a hydrologic context-aware gating mechanism that conditions expert outputs on seasonal phases, precipitation patterns, and basin attributes to address non-stationarity.
• The framework enhances label efficiency with semi-supervised multi-task training and optional large-model adaptation, achieving significant reductions in MSE and MAE relative to baselines.

Searching arXiv for the exact topic and closely related acronym usages to ground the article in current papers. {"query":"HydroFusion-LMF arXiv (Fan et al., 4 Oct 2025)"} HydroFusion-LMF is a unified framework for long-term daily runoff forecasting that combines learnable decomposition, heterogeneous expert fusion, hydrologic context-aware gating, semi-supervised multi-task training, and optional parameter-efficient adaptation of a frozen foundation time-series encoder. It was introduced for the setting of accurate decade-scale daily runoff forecasting in small watersheds, where signals blend drifting trends, multi-scale seasonal cycles, regime shifts, and sparse extremes. The framework performs a learnable trend-seasonal-residual decomposition, routes residuals through a compact heterogeneous expert set, fuses expert outputs via a hydrologic context-aware gate conditioned on day-of-year phase, antecedent precipitation, local variance, flood indicators, and static basin attributes, and augments supervision with a semi-supervised multi-task objective. On a ~10-year daily dataset it attains MSE 1.0128 / MAE 0.5818, improving the strongest baseline (DLinear) by 10.2% / 10.3% and the mean baseline by 24.6% / 17.1% (Fan et al., 4 Oct 2025).

1. Forecasting problem and design rationale

HydroFusion-LMF is motivated by the difficulty of long-horizon hydrologic prediction under strong non-stationarity. The target regime is daily runoff forecasting in small watersheds over long observational spans, where slow drift, multi-scale seasonality, low-frequency regime shifts, and infrequent extremes coexist. The underlying problem is not only predictive but also statistical: labeled supervision is scarce and variable, and prior deep models frequently target isolated structural biases while under-utilizing unlabeled spans (Fan et al., 4 Oct 2025).

The framework is positioned against a baseline set that includes DLinear, TimesNet, PatchTST, TiDE, Nonstationary Transformer, LSTNet, and LSTM. These models are described as usually targeting single facets, which limits regime adaptivity in the presence of structural change and sparse extremes. HydroFusion-LMF therefore assembles multiple inductive biases in a modular architecture rather than committing to a single forecasting mechanism. This suggests a deliberate response to hydrologic non-stationarity: trend, seasonality, local memory, long-range dependence, and regime-conditioned fusion are treated as distinct modeling problems rather than as side effects of one generic backbone (Fan et al., 4 Oct 2025).

A central theme is label efficiency. The method is designed to exploit unlabeled periods through masked reconstruction, multi-scale contrastive alignment, augmentation consistency, and variance-filtered pseudo-labeling, while retaining hydrologically meaningful supervised criteria such as MSE, MAE, NSE, and KGE. In this sense, HydroFusion-LMF is both a forecasting architecture and a training framework (Fan et al., 4 Oct 2025).

2. Learnable decomposition and heterogeneous expert ensemble

The architectural core begins with a learnable trend-seasonal-residual decomposition. For an observed series segment, the model partitions the signal as

$x_t = T_t + S_t + R_t .$

The trend term is a learned linear projection across lags,

$T_t = \mathbf{w}^\top \mathbf{x}_{t-L+1:t} + b ,$

and the seasonal term is represented by a truncated, learnable Fourier-like basis,

$$S_t = \sum_{m=1}^{M_f} \alpha_m \cos\left(\frac{2\pi m t}{\tau}\right) + \beta_m \sin\left(\frac{2\pi m t}{\tau}\right) .$$

The residual is then defined as

$R_t = x_t - T_t - S_t .$

In the paper’s formulation, the seasonal coefficients are emitted by a compact convolutional spectral encoder, allowing adaptive seasonality rather than a fixed harmonic template. Unlike fixed filter decompositions such as STL, the decomposer is end-to-end learnable and is described as adapting to evolving climate and anthropogenic modulations while cooperating with downstream modules during training (Fan et al., 4 Oct 2025).

After decomposition, HydroFusion-LMF routes residuals through a compact heterogeneous expert set. The expert types are: linear refinement, frequency kernel, patch Transformer, local recurrent memory, and dynamically normalized attention. Each expert is intended to capture a different residual structure: linear refinement handles smoothing and fine baseline adjustments; the frequency kernel addresses leaked or unmodeled periodic energy; the patch Transformer models long-range, regime-detecting dependencies; the LSTM-based local recurrent memory emphasizes short-term autoregressive behavior; and dynamically normalized attention combats scale and range drift through local recentering. The paper characterizes this as a compact mixture-of-experts design that enables contextual specialization without parameter explosion (Fan et al., 4 Oct 2025).

The residual forecast is formed as a convex combination of expert outputs,

$$\hat{r}_{t+h} = \sum_{k=1}^{K} g_t^{(k)} y_{t+h}^{(k)} .$$

This decomposition-plus-expert structure is the principal mechanism by which HydroFusion-LMF addresses non-stationarity: trend and seasonality absorb slow and quasi-periodic variation, while the expert ensemble concentrates on the more irregular component (Fan et al., 4 Oct 2025).

3. Hydrologic context-aware fusion and regime adaptivity

The fusion stage is organized around a hydrologic context-aware gating mechanism. The context vector includes day-of-year phase, antecedent precipitation index, local variance, rain percentile, a flood indicator, and static basin descriptors. The gate is implemented as a two-layer MLP that transforms hydrologic context into expert scores, followed by a softmax:

$$g_t^{(k)} = \frac{\exp(\tilde{g}_t^{(k)})}{\sum_{j=1}^{K} \exp(\tilde{g}_t^{(j)})} .$$

The final forecast is then reconstructed by recombining the decomposed components:

$$\hat{x}_{t+h} = \hat{T}_{t+h} + \hat{S}_{t+h} + \hat{r}_{t+h} .$$

This gate is not a generic attention layer detached from domain semantics; it is explicitly conditioned on hydrologic regime descriptors. The paper presents this as a mechanism for regime adaptivity, with the gate selecting or emphasizing experts according to seasonal position, precipitation history, volatility, and potential flood state. Prior models are described as ignoring or only statically encoding such context, whereas HydroFusion-LMF uses it directly in the fusion rule (Fan et al., 4 Oct 2025).

Interpretability is tied to this gating structure. Gating sparsity and entropy regularization are used so that the dominant expert under a given regime can be inspected. The paper gives illustrative roles: attention expert dominance during transitions, linear refiners during recession, and frequency kernels during stable seasonality. This is interpretability in the sense of explicit components and sparse routing, rather than in the stronger sense of causal attribution (Fan et al., 4 Oct 2025).

4. Semi-supervised objective and large-model adaptation

HydroFusion-LMF uses a composite objective that combines supervised, self-supervised, and semi-supervised terms:

$$\mathcal{L}_{\text{total}} = \lambda_{\text{sup}}\mathcal{L}_{\text{sup}} + \lambda_{\text{mask}}\mathcal{L}_{\text{mask}} + \lambda_{\text{ctr}}\mathcal{L}_{\text{ctr}} + \lambda_{\text{cons}}\mathcal{L}_{\text{cons}} + \lambda_{\text{pl}}\mathcal{L}_{\text{pl}} + \lambda_{\text{reg}}\mathcal{L}_{\text{reg}} .$$

The supervised term is itself composite:

$$\mathcal{L}_{\text{sup}} = \gamma_1 \mathcal{L}_{\text{mse-mae}} + \gamma_2 \mathcal{L}_{\text{ext}} + \gamma_3 \mathcal{L}_{\text{nse}} + \gamma_4 \mathcal{L}_{\text{kge}} .$$

Here, the framework couples pointwise losses with hydrologic metrics and an extreme-event emphasis term. This choice is notable because the training signal is aligned not only with generic regression accuracy but also with hydrology-specific evaluation criteria. The semi/self-supervised terms include masked reconstruction, multi-scale contrastive alignment using InfoNCE across daily, weekly, and monthly aggregations, augmentation consistency, and variance-filtered pseudo-labeling based on low-variance expert consensus. Regularization includes gating entropy and weight decay (Fan et al., 4 Oct 2025).

An additional feature is optional adapter / LoRA-based foundation time-series encoder injection. A frozen encoder $F_{\text{pre}}$ processes the input, and a lightweight adapter or LoRA subnetwork adds a domain-specific correction:

$T_t = \mathbf{w}^\top \mathbf{x}_{t-L+1:t} + b ,$0

This mechanism is presented as a way to import generic temporal invariances from pre-trained masked time-series Transformers while preserving parameter efficiency and avoiding destabilization of optimization. In the terminology of the paper’s title, this is the “Large-Model Adaptation” component of HydroFusion-LMF (Fan et al., 4 Oct 2025).

5. Empirical performance, extreme events, and ablation evidence

The empirical study reports multi-horizon aggregated MSE and MAE for HydroFusion-LMF and a set of forecasting baselines. The reported values are as follows (Fan et al., 4 Oct 2025):

Model	MSE	MAE
PatchTST	1.215	0.653
DLinear	1.128	0.649
TimesNet	1.332	0.702
LSTNet	1.299	0.693
Nonstationary Transformer	1.315	0.785
TiDE	1.225	0.673
LSTM	1.888	0.758
HydroFusion-LMF	1.013	0.582

In the abstract, the framework is reported to attain MSE 1.0128 / MAE 0.5818, improving the strongest baseline, DLinear, by 10.2% / 10.3%, and the mean baseline by 24.6% / 17.1%. The paper explicitly notes simultaneous MSE and MAE reductions relative to baselines, which is presented as a distinguishing outcome of the full model (Fan et al., 4 Oct 2025).

Performance on extreme events is also highlighted. HydroFusion-LMF is reported to achieve lower peak discharge error, at 2.73 versus 3.45+, improved timing, and the highest high-flow F1 score, at 0.82 versus 0.74 or below. These results are consistent with the inclusion of extreme-event emphasis in the supervised loss and with the use of hydrologic context features such as flood indicators in the gate (Fan et al., 4 Oct 2025).

The ablation study attributes the largest degradation to removing the learnable trend-seasonal-residual decomposition, which causes +14% MSE. Replacing the gating mechanism with uniform averaging causes +7% MSE. The paper further states that the semi-supervised loss and each expert type contribute significant, complementary value. This places the decomposition and the context-aware fusion mechanism at the center of the architecture’s empirical gains (Fan et al., 4 Oct 2025).

6. Interpretability, scope, and acronym disambiguation

HydroFusion-LMF is described as balancing interpretability with performance. Its interpretability derives from explicit decomposition into trend, seasonal, and residual components; sparse, context-driven gating; and semantically differentiated experts whose roles can be inspected under different hydrologic regimes. Its handling of non-stationarity similarly follows from multiple coordinated mechanisms: decomposition absorbs drift and phase or amplitude changes, experts and gating adapt to local regime, and semi-supervised learning regularizes representations on data-poor or structurally novel intervals (Fan et al., 4 Oct 2025).

A common source of confusion is the acronym “LMF.” In HydroFusion-LMF, the title expands LMF as “Multi-Network Fusion with Large-Model Adaptation.” This usage is distinct from “Low-rank Multimodal Fusion” in multimodal representation learning, which addresses tensor-based fusion efficiency through low-rank modality-specific factors (Liu et al., 2018). It is also distinct from “Local Molecular Field” theory in statistical mechanics and molecular simulation, where LMF denotes a framework for separating short- and long-range interactions in uniform polar molecular liquids, hydrophobic interactions, and water nucleation (Rodgers et al., 2010, Chaudhari et al., 2015, Zhao et al., 2023). It is likewise unrelated to liquid metal fusion studies on thick liquid metal walls and full-induction blanket MHD, where “fusion” refers to magnetic-confinement reactor engineering rather than hydrologic forecasting (Giovacchini et al., 2024, Endeve et al., 19 Nov 2025).

Within hydrology, the significance of HydroFusion-LMF lies in the particular synthesis it proposes: learnable decomposition, expert specialization, hydrologic context-aware fusion, semi-supervised regularization, and optional foundation-model adaptation are all treated as coordinated components of one forecasting framework. This suggests a broader methodological direction for runoff prediction under non-stationarity: not a single universal inductive bias, but a structured combination of explicit components, sparse gating, and label-efficient training (Fan et al., 4 Oct 2025).