Zhang et al. (2026) A high-order Model-free Dynamic Framework for Accurate Daily Streamflow Prediction

Identification

• Journal: Water Resources Management
• Year: 2026
• Date: 2026-03-01
• Authors: Jiaying Zhang, Shiqian Tang, Longxia Qian, Mei Hong, Yong Zhao, Linlin Fan
• DOI: 10.1007/s11269-026-04515-4

Research Groups

• School of Science, Nanjing University of Posts and Telecommunications, Nanjing, China
• State Key Laboratory of Water Resources Engineering and Management, Wuhan University, Wuhan, China
• College of Meteorology and Oceanography, National University of Defense Technology, Changsha, China
• Key Laboratory of Water Safety for Beijing-Tianjin-Hebei Region of Ministry of Water Resources, Beijing, China
• Agricultural Water Conservancy Department, Changjiang River Scientific Research Institute, Wuhan, China

Short Summary

This paper introduces a high-order lightweight dynamic framework (HoLDF) for daily streamflow forecasting, which integrates high-order structural information identified by an improved Granger causality inference approach into a reservoir computing paradigm. HoLDF significantly outperforms baseline deep learning models in accuracy, robustness, and computational efficiency, making it suitable for operational deployment.

Objective

• To develop a high-order lightweight dynamic framework (HoLDF) that integrates high-order structural information into a Reservoir Computing (RC) paradigm to achieve accurate and computationally efficient daily streamflow prediction.

Study Configuration

• Spatial Scale: Natural catchments in the northeastern United States (Maine and surrounding areas). Five hydrological stations were used, with four selected as target stations for prediction. Drainage areas range from 1.77826 x 10^9 square meters to 3.67617 x 10^9 square meters.
• Temporal Scale: Daily streamflow data from January 1, 1980, to November 7, 2012 (12,000 days). The forecast period was a continuous 365-day daily streamflow sequence randomly sampled from the test set.

Methodology and Data

• Models used:
  • Proposed: High-order lightweight dynamic framework (HoLDF) based on Reservoir Computing (RC) with an improved Granger causality inference approach.
  • Baselines: Gated Recurrent Unit (GRU), Sequence-to-Sequence (Seq2Seq), Temporal Convolutional Network (TCN), Long Short-Term Memory (LSTM).
• Data sources: CAMELS dataset, specifically measured daily streamflow data from five stations (USGS01013500, USGS01030500, USGS01144000, USGS01543500, and USGS01548500).

Main Results

• HoLDF significantly reduced Root Mean Square Error (RMSE) values by 45.33% to 86.33% and improved Nash-Sutcliffe Efficiency (NSE) values by 0.233 to 0.575 compared to baseline models.
• The framework demonstrated superior performance in peak streamflow prediction, reducing the average Absolute Relative Error (ARE) by 69.72% to 80.55% compared to benchmarks and accurately capturing peak timing without temporal lag.
• HoLDF exhibited higher computational efficiency with fewer trainable parameters and shorter runtime; other models required more than twice the training parameters and runtime. Peak CPU memory usage was 1.08262 x 10^9 bytes during training and 1.00938 x 10^9 bytes during testing.
• HoLDF maintained robust performance under varying levels of Gaussian noise perturbations (e.g., standard deviation of 0.2) and with reduced sequence lengths (e.g., 8,000 days), indicating strong adaptability to suboptimal data quality and volume.
• Statistical tests (R², DM test, ANOVA, paired t-test) confirmed HoLDF's strong linear correlation between predictions and observations, significantly smaller errors than baselines, and ability to capture site-specific hydrological characteristics.

Contributions

• Development of a high-order lightweight dynamic framework (HoLDF) that offers low computational cost and high operational efficiency, compensating for deep learning constraints and advancing Reservoir Computing in the field of hydrology.
• Integration of high-order structural information into predictive models by identifying high-order interactions among streamflow processes across diverse catchments via an improved Granger causality approach, thereby enhancing Reservoir Computing forecasting accuracy.

Funding

• Humanities and Social Science Fund of Ministry of Education (Grant 23YJAZH111)
• Visiting Researcher Fund Program of State Key Laboratory of Water Resources Engineering and Management (Grant No. 2025SWG04)
• Natural Science Foundation of Hunan Province (2023JJ10054)
• National Natural Science Foundation of China (Grant Nos. 42375016)
• Open Research Fund of Key Laboratory of Water Safety for Beijing-Tianjin-Hebei Region of Ministry of Water Resources (Grant NO. IWHR-KLWS-202301)
• Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant NO. SJCX25_0379)

Citation

@article{Zhang2026highorder,
  author = {Zhang, Jiaying and Tang, Shiqian and Qian, Longxia and Hong, Mei and Zhao, Yong and Fan, Linlin},
  title = {A high-order Model-free Dynamic Framework for Accurate Daily Streamflow Prediction},
  journal = {Water Resources Management},
  year = {2026},
  doi = {10.1007/s11269-026-04515-4},
  url = {https://doi.org/10.1007/s11269-026-04515-4}
}