Gia et al. (2025) ThoR: A Motion-Dependent Physics-Informed Deep Learning Framework with Constraint-Centric Theory of Functional Connections for Rainfall Nowcasting

Identification

• Journal: Scientific Reports
• Year: 2025
• Date: 2025-11-26
• Authors: Khang Ta Gia, Hoat Nguyen Van, An Phan Thanh, N. Minh
• DOI: 10.1038/s41598-025-26126-6

Research Groups

• Faculty of Computer Science and Engineering, Ho Chi Minh City University of Technology (HCMUT), Vietnam
• Institute of Mathematical and Computational Sciences (IMACS), Ho Chi Minh City University of Technology (HCMUT), Vietnam
• Vietnam National University Ho Chi Minh City (VNU-HCM), Vietnam
• Faculty of Information Technology, University of Danang - University of Science and Technology, Da Nang, Vietnam

Short Summary

This paper introduces ThoR, a motion-dependent physics-informed deep learning framework with a Constraint-Centric Theory of Functional Connections (TFC) for rainfall nowcasting. ThoR integrates attention-centric spatiotemporal modeling with explicit physical constraints (advection-diffusion equation) to achieve superior deterministic precipitation forecasts, particularly for extreme weather events and longer lead times, outperforming existing methods on real-world radar datasets.

Objective

• To develop a novel physics-informed deep learning framework (ThoR) that integrates attention-centric spatiotemporal modeling with explicit physical constraints derived from partial differential equations (PDEs) for accurate, physically consistent, and interpretable precipitation nowcasting.

Study Configuration

• Spatial Scale:
  • MRMS dataset: Continental U.S., 1 km spatial resolution, analyzed in 256 × 256 pixel subregions.
  • Nha Be dataset: Ho Chi Minh City, Vietnam (geographical region from 7.950°N to 13.360°N and 103.960°E to 109.500°E), 500 m spatial resolution, 2305 × 2881 grid points, covering approximately a 120 km radius.
• Temporal Scale:
  • Data frequency: 10-minute intervals.
  • Input sequence: 6 past frames (representing 1 hour of historical data).
  • Forecast horizon: 12 future frames (predicting the next 2 hours).
  • Data period: MRMS (2016-2022), Nha Be (2023).

Methodology and Data

• Models used:
  • ThoR Framework: A Motion-Dependent Physics-Informed Deep Learning Framework with Constraint-Centric Theory of Functional Connections (TFC) for Rainfall Nowcasting.
    • Architecture: Cascaded-branch design integrating an attention-driven generator with an unsupervised, lead-time-conditioned module for motion field extraction.
    • Physical Constraints: Explicitly embeds the advection-diffusion equation into the optimization objective via a weighted loss, establishing a TFC framework.
    • Motion Estimation: Uses a Conditioned discretised U-Net for Operator Learning, regularized by Horn-Schunck smoothness constraints and temporal evolution modeled by 2D Burgers' equations.
    • Generator: Motion-Conditioned ConvGRU with axial self-attention blocks for spatial aggregation.
    • Adversarial Learning: Utilizes a Generative Adversarial Network (GAN) paradigm with a PatchGAN-based discriminator.
    • Loss Function: Composite objective including data fidelity (L1, L2, Huber, and Intensity Consistency Loss with a trinary cross-entropy mask), physics-consistency (advection-diffusion equation), motion consistency (Burgers' equation, Horn-Schunck), and adversarial loss.
  • Baseline Models for Comparison: PySTEPS, NowcastNet, DiffCast, PredRNN, TrajGRU.
• Data sources:
  • MRMS dataset: NOAA’s Multi-Radar Multi-Sensor system, providing radar-derived Precipitation Rate fields across the continental U.S. (2016-2022). Publicly available via HuggingFace.
  • Nha Be dataset: High-resolution radar reflectivity dataset from the Nha Be Weather Radar Station, Ho Chi Minh City, Vietnam (2023). Used under restricted license.
  • Ancillary data: GeoPandas library with Natural Earth dataset (county-level boundaries for U.S. map), Cartopy library with Natural Earth coastline data (for Nha Be map).

Main Results

• ThoR consistently outperforms five state-of-the-art baseline methods (PySTEPS, NowcastNet, DiffCast, PredRNN, TrajGRU) across all evaluation metrics (Mean Squared Error, Structural Similarity Index, Critical Success Index at 1, 8, and 16 mm/h thresholds) on both the MRMS and Nha Be datasets.
• Quantitative Performance:
  • Achieved an average Mean Squared Error (MSE) of 41.46, representing a 2.01% improvement over the strongest baseline (NowcastNet).
  • Attained an average Structural Similarity Index (SSIM) of 0.91, demonstrating robust preservation of fine-grained spatial patterns.
  • Showed superior performance in detecting moderate to heavy rainfall events, with an average CSI(8) of 0.599 and the highest average CSI(16) of 0.326, indicating enhanced capability in identifying high-intensity, localized precipitation cells crucial for flood risk assessment.
• Lead Time Robustness: ThoR consistently maintained superior performance in CSI and SSIM compared to baselines as lead time increased, highlighting its robustness for longer-term nowcasts (up to 2 hours).
• Qualitative Performance: Produced sharper forecasts with spectral characteristics more closely aligned with radar observations, demonstrating greater accuracy in predicting high rainfall intensities and reduced smoothing effects, particularly for extreme weather events and in tropical convective storm regimes (Nha Be dataset).
• Ablation Studies: Confirmed the critical role of explicit motion extraction, adversarial learning, and the integration of physics-informed and motion consistency losses in significantly improving predictive accuracy and physical consistency.

Contributions

• Extended the Theory of Functional Connections (TFC) by directly incorporating the advection-diffusion equation and associated physical constraints of the motion field into the optimization objectives, leading to a "Soft TFC with constraint-centric design" that enhances model interpretability and predictive performance for precipitation nowcasting.
• Proposed ThoR, a novel hybrid deep learning architecture that integrates attention-based spatiotemporal modeling with explicit physical constraints derived from partial differential equations (PDEs). ThoR accurately simulates precipitation dynamics, delivering high-quality forecasts across a broad spectrum of rainfall intensities, with particularly strong performance in predicting extreme precipitation events governed by advective and convective processes.

Funding

• Ministry of Natural Resources and Environment (MONRE), Vietnam (grant number TNMT.2024.06.06).
• Ho Chi Minh City University of Technology (HCMUT), VNU-HCM.

Citation

@article{Gia2025ThoR,
  author = {Gia, Khang Ta and Van, Hoat Nguyen and Thanh, An Phan and Minh, N.},
  title = {ThoR: A Motion-Dependent Physics-Informed Deep Learning Framework with Constraint-Centric Theory of Functional Connections for Rainfall Nowcasting},
  journal = {Scientific Reports},
  year = {2025},
  doi = {10.1038/s41598-025-26126-6},
  url = {https://doi.org/10.1038/s41598-025-26126-6}
}