Li et al. (2026) A physically based neural network for flood routing: The Muskingum-Recurrent neural network

Identification

• Journal: Journal of Hydrology
• Year: 2026
• Date: 2026-04-01
• Authors: Zhaoxi Li, Tiejian Li, Jian Sun, Jiaye Li, Weidong Li, Jie Zhao, Jiahua Wei
• DOI: 10.1016/j.jhydrol.2026.135403

Research Groups

• Department of Hydraulic Engineering, Tsinghua University, Beijing, China
• State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, China
• School of Environment and Civil Engineering, Dongguan University of Technology, Dongguan, China
• State Key Laboratory of Plateau Ecology and Agriculture, Qinghai University, Xining, China

Short Summary

This study develops a Muskingum-Recurrent Neural Network (MRNN) that integrates the Muskingum flood routing equations directly into the RNN architecture, enforcing mass conservation as a hard constraint. The MRNN demonstrates superior data efficiency, robustness, and physical consistency in flood routing compared to conventional neural networks and traditional process-based methods across artificial, benchmark, and real-world flood events.

Objective

• To develop a physically-based neural network (Muskingum-Recurrent Neural Network, MRNN) for flood routing by integrating the Muskingum routing equations into the Recurrent Neural Network (RNN) internal structure, with Muskingum coefficients defining the network's weights, thereby ensuring mass conservation as a hard architectural constraint.

Study Configuration

• Spatial Scale:
  • Artificial channels: Four types (rectangular, trapezoidal; one-segment 100 km, two-segment 200 km).
  • Classical benchmark floods: Four cases (Wilson, Wye River, Viessman and Lewis, Nanyun River) representing various river reaches.
  • Real-world basin: Village Creek watershed, Birmingham, Alabama, USA (total length 70.8 km, watershed area 103.6 km²).
• Temporal Scale:
  • Artificial channels: Total duration of 72 hours, routing time step of 10 minutes (600 seconds).
  • Real-world basin: Hourly streamflow data, flood events exceeding 15 hours duration. Data collected since 1997.

Methodology and Data

• Models used:
  • Proposed: Muskingum-Recurrent Neural Network (MRNN), derived from the Variable Parameter Muskingum Method (VPMM).
  • Benchmark Neural Networks: Artificial Neural Network (ANN), Recurrent Neural Network (RNN), Long Short-Term Memory (LSTM).
  • Benchmark Process-Based Muskingum Methods: Over ten optimization algorithms including Particle Swarm Optimization (PSO), Big Bang-Big Crunch Algorithm (BFGSA), Firefly Algorithm (FORK), Honey Bee Mating Optimization (HBMO), Cuckoo Search Algorithm (COBSA), Real-coded Genetic Algorithm with Nelder-Mead Simplex (RAGA-NMS), Teaching-Learning-Based Optimization (TLBO), and others.
• Data sources:
  • Artificial channels: 8000 synthetic flood hydrographs (1000 single-peak, 1000 double-peak for each of four channel types) generated using a one-dimensional river network hydrodynamic model.
  • Classical benchmark floods: Four widely used cases from literature (Wilson, Wye River, Viessman and Lewis, Nanyun River).
  • Observation data: Hourly streamflow data from U.S. Geological Survey (USGS) stations 02458600 and 02458502 in the Village Creek Basin, USA.

Main Results

• Convergence and Efficiency: MRNN converges significantly faster (approx. 10 epochs) than ANN (30 epochs), RNN (1000 epochs), and LSTM (3000 epochs). It achieves comparable or superior performance using substantially fewer parameters (18-36 parameters for MRNN vs. millions for conventional NNs).
• Overall Performance: MRNN demonstrates superior overall performance in artificial channels (higher Nash-Sutcliffe Efficiency (NSE), Kling-Gupta Efficiency (KGE), and lower Percent Bias (Pbias)) compared to ANN, RNN, and LSTM.
• Hydrological Accuracy: MRNN more accurately simulates flood peak discharge (Relative Error of Peak Discharge (REPD) reduced by 24.54%-29.12%) and peak timing (Absolute Error of Peak Time (AEPT) reduced by 12.72%-33.63%) in artificial channels.
• Mass Conservation: MRNN consistently exhibits orders of magnitude less water loss (average ΔS of 2.57×10³ m³ in artificial channels, 1.39×10³ m³ in Village Creek) compared to ANN, RNN, and LSTM (e.g., 500 to 3500 times smaller). This is due to mass conservation being a hard architectural constraint.
• Comparison with Process-Based Methods: Multi-layer MRNN (e.g., 3-layer) often achieves lower Mean Squared Error (MSE) and REPD than most traditional Muskingum parameter optimization methods, particularly for complex hydrographs (e.g., 34.9% MSE reduction for Wye River flood compared to the best traditional method). MRNN accurately captured peak timing (AEPT = 0 h) for complex cases where traditional methods showed errors.
• Robustness: MRNN is the least sensitive to training set size. Reducing the training set from 80% to 20% resulted in only a 2.81% decrease in NSE and 2.06% decrease in KGE for MRNN, significantly less degradation than other models.

Contributions

• Novel Architecture: Introduces a new paradigm for physics-informed neural networks by directly embedding the Variable Parameter Muskingum Method (VPMM) equations into the Recurrent Neural Network (RNN) architecture, establishing mathematical equivalence and defining network weights with physical coefficients.
• Hard Physical Constraint: Guarantees mass conservation as a hard architectural constraint, leading to significantly improved physical consistency and interpretability compared to soft-constraint (PINN) or modular hybrid approaches.
• Data Efficiency and Robustness: Achieves high performance with a minimal number of physically interpretable parameters, demonstrating superior data efficiency and robustness to limited training data due to a strong inductive bias.
• Computational Advantage: Exhibits faster convergence and reduced computational cost compared to conventional data-driven neural networks.
• Enhanced Interpretability: Each parameter (a, b, x) and internal component of the MRNN cell has a direct physical meaning, enhancing model interpretability.

Funding

• National Natural Science Foundation of China (U2243201)
• Major Science and Technology Project of Qinghai Province (2021-SF-A6)

Citation

@article{Li2026physically,
  author = {Li, Zhaoxi and Li, Tiejian and Sun, Jian and Li, Jiaye and Li, Weidong and Zhao, Jie and Wei, Jiahua},
  title = {A physically based neural network for flood routing: The Muskingum-Recurrent neural network},
  journal = {Journal of Hydrology},
  year = {2026},
  doi = {10.1016/j.jhydrol.2026.135403},
  url = {https://doi.org/10.1016/j.jhydrol.2026.135403}
}