Introduction

Microplastic pollution has emerged as one of the most pressing environmental challenges of the 21st century, with riverine systems serving as critical pathways for the transport of these persistent pollutants from terrestrial sources to marine environments1. The ubiquitous presence of microplastics in river ecosystems worldwide has been documented extensively, with concentrations ranging from hundreds to millions of particles per cubic meter, depending on local pollution sources and hydrodynamic conditions2. These microscopic plastic fragments, typically defined as particles smaller than 5 mm, pose significant ecological risks due to their ability to adsorb toxic chemicals, interact with aquatic organisms, and persist in the environment for extended periods3.

The transport and fate of microplastics in riverine environments are governed by complex hydrodynamic processes that operate across multiple spatial and temporal scales4. River flow dynamics, including velocity patterns, turbulence characteristics, and sediment transport mechanisms, play crucial roles in determining microplastic distribution, accumulation, and downstream migration5. The interaction between microplastic properties—such as size, density, and shape—and hydrodynamic forces creates intricate transport patterns that are challenging to predict using conventional modeling approaches6. Furthermore, the temporal variability of river discharge, seasonal flow variations, and extreme events introduce additional complexity to microplastic transport processes7.

Current research methodologies for investigating microplastic transport in riverine systems predominantly rely on field monitoring, laboratory experiments, and traditional numerical modeling approaches8. While these methods have provided valuable insights into microplastic behavior, they often suffer from limitations in capturing the full complexity of spatiotemporal transport processes. Field studies are constrained by sampling frequency and spatial coverage, while laboratory experiments may not adequately represent real-world conditions9. Traditional numerical models, although useful for understanding fundamental transport mechanisms, typically struggle to incorporate the non-linear interactions between multiple environmental factors and the heterogeneous nature of river systems10.

The advent of spatiotemporal graph neural networks presents unprecedented opportunities for advancing our understanding of microplastic transport mechanisms in riverine environments. These advanced machine learning architectures can effectively capture complex spatial relationships between different river segments while simultaneously modeling temporal dependencies in hydrodynamic processes. By leveraging the graph-based representation of river networks and incorporating temporal dynamics, spatiotemporal graph neural networks offer superior capabilities for analyzing the intricate relationships between hydrodynamic variables and microplastic transport patterns compared to conventional approaches.

The primary objective of this research is to develop and implement a comprehensive spatiotemporal graph neural network framework for elucidating the influence mechanisms of river hydrodynamics on microplastic transport processes. This study aims to establish a novel analytical approach that can effectively integrate multi-scale hydrodynamic data with microplastic transport observations to reveal underlying transport mechanisms and improve predictive capabilities. The research significance lies in its potential to provide new insights into microplastic pollution dynamics in riverine systems and support the development of more effective pollution control strategies. The technical approach involves constructing a graph-based representation of river networks, incorporating temporal hydrodynamic variables, and developing specialized neural network architectures capable of capturing the complex spatiotemporal relationships governing microplastic transport in riverine environments.

Literature review and theoretical foundation

Transport mechanisms of microplastics in river environments

The transport behavior of microplastics in riverine systems is fundamentally governed by their physicochemical properties, which determine the complex interactions between particles and hydrodynamic forces11. Particle density represents one of the most critical factors influencing microplastic mobility, with buoyant particles (ρ_p < ρ_w) exhibiting different transport patterns compared to dense particles that tend to settle toward the riverbed12. The shape factor and surface characteristics of microplastics significantly affect their drag coefficients and settling velocities, with irregular fragments demonstrating more complex transport behaviors than spherical particles due to enhanced surface area and altered hydrodynamic resistance13.

The settling velocity of microplastic particles in river flows can be described by Stokes’ law for small Reynolds numbers:

$$\:{w}_{s}=\frac{2g\left({\rho\:}_{p}-{\rho\:}_{w}\right){d}^{2}}{18\mu\:}$$

where w_s is the settling velocity, g is gravitational acceleration, ρ_p and ρ_w are particle and water densities respectively, d is the particle diameter, and µ is the dynamic viscosity of water14. However, for larger particles or higher flow velocities, the drag force becomes significant, requiring the consideration of the drag coefficient:

$$\:{w}_{s}=\sqrt{\frac{4g\left({\rho\:}_{p}-{\rho\:}_{w}\right)d}{3{C}_{D}{\rho\:}_{w}}}$$

where C_D is the drag coefficient that varies with particle Reynolds number and shape characteristics15.

The dynamic processes of particle settling, suspension, and resuspension in river environments are controlled by the balance between gravitational forces and turbulent shear stresses. The critical shear stress for particle resuspension can be expressed through the Shields parameter:

$$\:{\theta\:}_{c}=\frac{{\tau\:}_{c}}{{\rho\:}_{w}g\left({\rho\:}_{p}-{\rho\:}_{w}\right)d}$$

where θ_c is the critical Shields parameter, τ_c is the critical bed shear stress, and the denominator represents the submerged weight of the particle per unit area16. This relationship demonstrates that smaller and less dense microplastic particles require lower shear stresses for mobilization compared to larger or denser particles.

The interaction between microplastics and riverbed sediments introduces additional complexity to transport mechanisms, as particles can become temporarily trapped within sediment matrices or form aggregates that alter their effective density and size. Microplastic particles with densities close to water exhibit prolonged suspension times and enhanced downstream transport potential, while denser particles tend to accumulate in depositional zones such as channel bends, backwater areas, and confluence regions. The differential transport behavior based on particle size reveals that smaller microplastics (< 100 μm) remain predominantly in suspension and follow water mass movements, whereas larger particles (> 1 mm) exhibit saltation and bedload transport patterns similar to natural sediments. These size-dependent transport mechanisms result in spatial segregation of microplastic populations within river systems, with fine particles concentrated in the water column and coarser particles accumulating in benthic environments.

Fundamental theory of river hydrodynamics

The mathematical description of river hydrodynamics is founded upon the fundamental conservation equations that govern fluid motion and mass transport in natural watercourses17. The continuity equation represents the principle of mass conservation for incompressible flow:

$$\:\frac{\partial\:u}{\partial\:x}+\frac{\partial\:v}{\partial\:y}+\frac{\partial\:w}{\partial\:z}=0$$

where u, v, and w are velocity components in the x, y, and z directions respectively18. This equation establishes the constraint that the net flow of water into any control volume must equal the net flow out, providing the foundation for understanding flow field characteristics that directly influence microplastic transport patterns.

The momentum conservation equations, commonly known as the Navier-Stokes equations, describe the relationship between fluid acceleration, pressure gradients, and viscous forces in river flows19. For the streamwise direction, the momentum equation can be expressed as:

$$\:\frac{\partial\:u}{\partial\:t}+u\frac{\partial\:u}{\partial\:x}+v\frac{\partial\:u}{\partial\:y}+w\frac{\partial\:u}{\partial\:z}=-\frac{1}{\rho\:}\frac{\partial\:p}{\partial\:x}+\nu\:{\nabla\:}^{2}u+{g}_{x}$$

where p is pressure, ρ is fluid density, ν is kinematic viscosity, and g_x represents gravitational acceleration in the x-direction20. The left-hand side represents material acceleration, while the right-hand side accounts for pressure forces, viscous forces, and gravitational effects that collectively determine the velocity field governing microplastic advection.

The transport equation for suspended particles, including microplastics, incorporates both advective and diffusive transport mechanisms:

$$\:\frac{\partial\:C}{\partial\:t}+\frac{\partial\:\left(uC\right)}{\partial\:x}+\frac{\partial\:\left(vC\right)}{\partial\:y}+\frac{\partial\:\left(wC\right)}{\partial\:z}=\frac{\partial\:}{\partial\:x}\left({D}_{x}\frac{\partial\:C}{\partial\:x}\right)+\frac{\partial\:}{\partial\:y}\left({D}_{y}\frac{\partial\:C}{\partial\:y}\right)+\frac{\partial\:}{\partial\:z}\left({D}_{z}\frac{\partial\:C}{\partial\:z}\right)+S$$

where C represents particle concentration, D_x, D_y, and D_z are dispersion coefficients in respective directions, and S represents source and sink terms21.

The velocity distribution in river channels exhibits characteristic logarithmic profiles in the vertical direction, with maximum velocities occurring near the water surface and decreasing toward the bed due to boundary friction effects. This vertical velocity gradient creates differential transport conditions for microplastics at various depths, with particles in surface waters experiencing higher velocities and enhanced downstream transport rates compared to those near the riverbed. Turbulent flow characteristics, quantified through parameters such as turbulent kinetic energy and Reynolds stresses, play crucial roles in maintaining particle suspension and promoting vertical mixing processes that prevent settling of buoyant microplastics.

The bed shear stress distribution across river cross-sections determines the spatial variability of particle resuspension potential, with higher shear stresses in channel thalwegs promoting mobilization of deposited microplastics, while lower stresses in shallow areas and backwater zones favor particle deposition22. The theoretical linkage between hydrodynamic parameters and microplastic transport processes reveals that flow velocity magnitude controls advective transport rates, while turbulence intensity governs vertical mixing and suspension maintenance. Additionally, the temporal variability of hydrodynamic conditions, particularly during flood events, significantly alters microplastic transport capacity and redistribution patterns within river systems, highlighting the importance of incorporating time-dependent flow characteristics in transport modeling frameworks.

Applications of graph neural networks in Spatiotemporal modeling

Graph neural networks represent a paradigm shift in machine learning architectures, specifically designed to process data structured as graphs where entities are connected through complex relationships23. The fundamental principle of GNNs lies in their ability to perform message passing between nodes, allowing information aggregation from neighboring nodes to update node representations iteratively. The basic GNN framework operates through node embedding updates that capture both local neighborhood information and global graph structure, making it particularly suitable for modeling systems with inherent network topology such as river systems24.

Spatiotemporal graph neural networks extend traditional GNN architectures by incorporating temporal dependencies alongside spatial relationships, addressing the challenge of modeling dynamic systems that evolve over time. These networks typically integrate convolutional operations over graph structures with recurrent or attention mechanisms to capture temporal patterns, enabling simultaneous modeling of spatial correlations and temporal dynamics25. The architecture commonly employs graph convolutional layers for spatial feature extraction followed by temporal modeling components such as gated recurrent units or temporal attention mechanisms, providing comprehensive representation of spatiotemporal phenomena.

The application of graph neural networks in environmental science has demonstrated remarkable success in various domains, including air quality prediction, hydrological modeling, and climate pattern analysis26. In fluid mechanics applications, GNNs have been successfully employed for turbulence modeling, flow field prediction, and computational fluid dynamics acceleration, where the ability to capture complex spatial relationships between fluid elements provides significant advantages over traditional grid-based approaches. These applications have highlighted the superior performance of GNNs in handling irregular geometries, adaptive mesh structures, and multi-scale phenomena that are characteristic of natural fluid systems27.

The suitability of graph neural networks for river system modeling stems from their inherent ability to represent river networks as graph structures, where river segments serve as nodes and hydraulic connections form edges. This representation naturally captures the topological relationships between different river reaches while enabling the incorporation of spatially varying hydrodynamic properties and boundary conditions. The graph-based approach facilitates the modeling of complex river networks with tributaries, confluences, and varying channel geometries that are difficult to represent using traditional numerical methods28.

The development prospects for spatiotemporal graph neural networks in river system applications appear highly promising, particularly for addressing challenges related to multi-scale transport processes, real-time prediction, and uncertainty quantification. Recent advances in attention mechanisms, graph transformer architectures, and physics-informed neural networks provide opportunities for incorporating physical constraints and domain knowledge into GNN frameworks. The integration of graph neural networks with traditional computational fluid dynamics approaches offers potential for developing hybrid modeling systems that combine the flexibility of machine learning with the reliability of physics-based models, ultimately advancing our capability to understand and predict complex environmental processes in riverine systems.

Spatiotemporal graph neural network model construction and methodology

Graph structure modeling of river systems

The representation of river systems as graph structures requires careful consideration of the spatial discretization strategy and topological relationships that govern hydrodynamic processes and microplastic transport29. Each river cross-section or computational reach is designated as a node in the graph, with the node positioning determined by the characteristic length scales of the transport phenomena and the spatial resolution requirements for capturing relevant hydrodynamic gradients. The node representation method adopts a hierarchical approach where primary nodes correspond to main channel reaches, while secondary nodes represent tributary confluences, hydraulic control structures, and monitoring stations that provide critical boundary conditions for the transport modeling framework30.

The graph structure construction emphasizes the preservation of river network connectivity and topological relationships that are essential for accurate representation of flow routing and mass transport processes. Directed edges connect upstream and downstream nodes following the natural flow direction, while the graph topology explicitly incorporates tributary junctions, channel bifurcations, and hydraulic control points that significantly influence flow patterns and microplastic distribution31. The connectivity matrix formulation ensures that the fundamental principles of mass conservation and flow continuity are maintained within the graph representation, providing a physically consistent framework for subsequent neural network operations. Figure 1 illustrates the comparison between traditional river modeling approaches and the proposed graph-based representation.

Fig. 1
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Comparison of traditional river modeling approach and graph-based representation. (a) Conventional one-dimensional river discretization showing uniform reach segments with fixed computational nodes. (b) Graph neural network representation with nodes as adaptive river reaches and edges representing hydraulic connectivity, enabling flexible topology and enhanced information flow.

The graph structure maintains physical consistency through strict adherence to mass conservation principles, where the connectivity matrix A ensures that \(\:\sum\:{A}_{\left\{ij\right\}}=\:1\) for each node i, preserving flow continuity across the network. The directed edge weights \(\:{w}_{\left\{ij\right\}}\) are formulated based on hydraulic residence time and momentum transfer coefficients, ensuring that information propagation respects the physical constraints of advective transport. The topological representation naturally captures the hierarchical structure of river networks while maintaining the Strahler ordering that governs flow accumulation and energy dissipation patterns in natural systems.

The node feature vector design encompasses comprehensive hydrodynamic parameters, geometric characteristics, and environmental factors that collectively determine microplastic transport behavior within each river reach.

Hydrodynamic parameters include flow velocity, water depth, discharge, bed shear stress, and turbulence intensity, which directly control particle advection, settling, and resuspension processes. Geometric features incorporate channel width, depth, slope, roughness coefficient, and cross-sectional area that influence flow resistance and velocity distribution patterns. Environmental factors encompass water temperature, suspended sediment concentration, and local pollution sources that affect microplastic properties and transport mechanisms32.

The edge weight calculation methodology quantifies the strength of hydrodynamic coupling between adjacent river reaches, reflecting the degree of influence that upstream conditions exert on downstream transport processes. The edge weights are formulated based on hydraulic travel time, flow velocity magnitude, and geometric transition characteristics between connected nodes. The weighting function incorporates dimensionless parameters such as Froude number and Reynolds number to capture the relative importance of inertial, gravitational, and viscous forces in governing inter-node transport processes. Additionally, the edge weights adapt dynamically to temporal variations in flow conditions, ensuring that the graph structure accurately represents changing hydrodynamic influences during different flow regimes, from low-flow periods that promote particle settling to high-flow events that enhance particle mobilization and downstream transport. This adaptive weighting mechanism enables the graph neural network to respond appropriately to temporal variations in river hydrodynamics while maintaining the underlying physical relationships that govern microplastic transport in riverine environments.

Spatiotemporal feature extraction and fusion mechanisms

The multi-scale spatiotemporal convolutional module design addresses the inherent multi-scale nature of microplastic transport processes, which operate simultaneously across different temporal and spatial scales ranging from turbulent eddies to seasonal flow variations33. Table 1 presents the configuration parameters for various spatiotemporal feature extraction components. The architecture employs parallel convolutional branches with varying kernel sizes and dilation rates to capture both short-term fluctuations in hydrodynamic conditions and long-term transport patterns. The spatial convolution operations leverage graph-based convolution kernels that aggregate information from neighboring nodes while preserving the topological structure of the river network, enabling effective extraction of spatial correlation patterns that reflect upstream-downstream transport relationships and lateral exchange processes.

Table 1 Configuration parameters for Spatiotemporal feature extraction components.
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$$\:{\alpha\:}_{t,s}=softmax\left({Q}_{t}{K}_{s}^{T}/\sqrt{{d}_{k}}\right)$$

where \(\:{Q}_{t}\) represents the query vector for temporal features and \(\:{K}_{s}\) represents the key vector for spatial features. The adaptive weight allocation follows the physics-informed principle that emphasizes advective processes during high-flow periods (α_spatial > 0.7) and local settling processes during low-flow conditions (α_temporal > 0.6), automatically adjusting based on the Froude number \(\:Fr=U/\sqrt{\left(gh\right)}\) as a regime indicator.

The temporal attention module employs scaled dot-product attention to compute attention weights that highlight critical temporal periods, such as flood events or seasonal transitions, that significantly influence transport patterns. The multi-head attention architecture enables the model to simultaneously focus on different temporal aspects of the transport process, with separate attention heads specializing in capturing short-term turbulent fluctuations, medium-term flow variations, and long-term seasonal patterns that collectively determine microplastic distribution dynamics.

The adaptive weight allocation strategy optimizes the balance between spatial neighborhood information and temporal sequence data through a learnable gating mechanism that dynamically adjusts the contribution of spatial and temporal features based on local hydrodynamic conditions35. Figure 2 demonstrates the comparison of spatiotemporal feature extraction effectiveness across different temporal scales. The spatial weight component emphasizes upstream influence during high-flow conditions when advective transport dominates, while increasing the temporal weight contribution during low-flow periods when local settling and resuspension processes become more significant. This adaptive mechanism incorporates hydrodynamic regime indicators such as Froude number and flow stability metrics to automatically adjust the spatial-temporal balance without manual parameter tuning.

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Spatiotemporal feature extraction effectiveness comparison. (a) Short-term feature extraction performance (hourly to daily scales) showing R² = 0.923 for capturing turbulent transport mechanisms. (b) Long-term pattern recognition (seasonal scales) demonstrating R² = 0.847 for seasonal variation identification with attention weight distribution.

The feature representation optimization employs dimensionality reduction techniques combined with learnable embedding layers to create compact yet informative feature encodings that capture the essential characteristics of microplastic transport processes36. The encoding strategy utilizes variational autoencoders to learn low-dimensional representations of high-dimensional hydrodynamic state vectors while preserving critical transport-relevant information. The feature dimension selection balances model complexity with representation capacity, employing information-theoretic criteria to determine optimal encoding dimensions that maximize transport prediction accuracy while minimizing computational overhead. The resulting feature representations enable efficient processing of large-scale river network data while maintaining the fidelity necessary for accurate microplastic transport modeling across diverse hydrodynamic conditions and temporal scales.

Microplastic transport prediction model design

The end-to-end spatiotemporal graph neural network prediction framework integrates multi-scale feature extraction modules with physics-informed prediction layers to establish a comprehensive modeling system for microplastic transport forecasting37. Table 2 presents the hyperparameter optimization results for the microplastic transport prediction model. The framework architecture consists of sequential processing stages including input data preprocessing, spatiotemporal feature encoding, graph-based information propagation, and multi-step prediction generation. The encoder-decoder structure enables the model to capture complex spatiotemporal dependencies while maintaining computational efficiency for real-time prediction applications in large-scale river networks.

Table 2 Hyperparameter optimization results for the microplastic transport prediction model.
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The loss function design incorporates both prediction accuracy objectives and physical constraints to ensure that the learned model respects fundamental transport principles38. The composite loss function is formulated as:

$$\:{L}_{total}=\:{L}_{prediction}+\:{L}_{physics}$$

where \(\:{L}_{prediction}=\:MSE\left({C}_{pred},\:{C}_{obs}\right)\) and the physics-informed loss L_physics comprises three constraint terms:

$$\:{L}_{physics}=\:{\lambda\:}^{1}{L}_{mass}+\:{\lambda\:}^{2}{L}_{velocity}+\:{\lambda\:}^{3}{L}_{boundary}$$

The mass conservation constraint enforces the advection-diffusion equation:

$$\:L_{{mass}} = \left( {\frac{1}{N}} \right)\sum\nolimits_{{\text{i}}} {\left| {\left| {\nabla \cdot \left( {u_{{\text{i}}} C_{{\text{i}}} } \right) + \frac{{\partial C_{{\text{i}}} }}{{\partial t}} - D\nabla ^{2} C_{{\text{i}}} } \right|} \right|^{2} }$$

The velocity boundary constraint ensures no-flux conditions at solid boundaries:

$$\:L_{{velocity}} = \left( {\frac{1}{M}} \right)\sum\nolimits_{{\text{j}}} {\left| {\left| {u_{{\text{j}}} \cdot n_{{\text{j}}} } \right|_{{\{ \partial \Omega \} }} } \right|^{2} }$$

The concentration boundary constraint maintains specified inlet/outlet conditions:

$$\:L_{{boundary}} = \left( {\frac{1}{K}} \right)\sum\nolimits_{{\text{k}}} {\left| {\left| {C_{{\text{k}}} - C_{{\left\{ {bc,k} \right\}}} } \right|_{{\left\{ {\partial \Omega } \right\}}} } \right|^{2} }$$

Weight parameters are determined through Bayesian optimization with \(\:{\lambda\:}^{1}=\:0.1\:\pm\:\:0.02\) for mass conservation (primary constraint), \(\:{\lambda\:}^{2}=\:0.05\:\pm\:\:0.01\) for velocity boundaries, and \(\:{\lambda\:}^{3}=\:0.15\:\pm\:\:0.03\) for concentration boundaries. The weights are adaptively scaled during training using the relative magnitude criterion: \(\:\lambda _{{\text{i}}} \left( {epoch} \right) = \lambda _{{\text{i}}}^{0} \times \left( {\frac{{L_{{prediction}} }}{{L_{{\text{i}}} }}} \right)^{\alpha }\) with α = 0.3 to maintain balanced constraint enforcement throughout optimization.

The physics-informed component includes regularization terms based on the advection-diffusion equation residuals, ensuring that predicted concentration fields satisfy the underlying transport physics within acceptable tolerance limits.

The model training strategy employs advanced techniques including data augmentation, regularization, and adaptive optimization algorithms to enhance generalization performance and prevent overfitting39. Figure 3 shows the comparison of prediction accuracy for microplastic concentration forecasting using different modeling approaches. Data augmentation techniques generate synthetic training samples through controlled perturbation of hydrodynamic conditions and microplastic properties, expanding the training dataset diversity while maintaining physical realism. The augmentation process includes temporal shifting, noise injection, and flow condition interpolation to create robust training scenarios that improve model performance under varying environmental conditions.

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Performance comparison of modeling approaches. (a) Prediction accuracy comparison showing spatiotemporal GNN achieving R² = 0.897 versus traditional numerical model R² = 0.643. (b) Computational efficiency demonstrating 92% reduction in processing time (12.4s vs. 158.3s) while maintaining superior accuracy.

The regularization framework combines L2 weight penalties with dropout mechanisms and early stopping criteria to maintain model generalizability across different river systems and hydrodynamic regimes. The adaptive optimization algorithm utilizes Adam optimizer with learning rate scheduling and gradient clipping to ensure stable convergence during training. The learning rate schedule incorporates warm-up periods and exponential decay to balance exploration and exploitation during the optimization process, preventing convergence to poor local minima while maintaining training stability.

The model validation and performance evaluation framework establishes comprehensive metrics for assessing prediction accuracy, physical consistency, and computational efficiency40. Cross-validation procedures employ temporal splitting strategies that respect the chronological order of data to prevent information leakage and ensure realistic performance assessment. Performance metrics include root mean square error, mean absolute percentage error, correlation coefficients, and Nash-Sutcliffe efficiency for quantitative accuracy assessment. Additionally, physics-based validation metrics evaluate mass balance conservation, particle settling behavior consistency, and boundary condition adherence to ensure that predictions remain physically meaningful across different scales and conditions. The evaluation framework also incorporates uncertainty quantification through ensemble prediction methods and confidence interval estimation to provide reliable uncertainty bounds for decision-making applications.

Results analysis and discussion

Model performance validation and comparative analysis

The validation of the spatiotemporal graph neural network model using field-measured microplastic concentration data from multiple river monitoring stations demonstrates superior predictive performance compared to conventional approaches41. Table 3 provides a comprehensive comparative performance evaluation of different modeling approaches for microplastic transport prediction. The model achieved a coefficient of determination (R²) of 0.897 for microplastic concentration predictions across different temporal scales, significantly outperforming traditional numerical models which typically achieve R² values between 0.6 and 0.7 for similar transport phenomena. The root mean square error (RMSE) was calculated using the following formula:

$$\:RMSE=\sqrt{\frac{1}{n}\sum\:_{i=1}^{n}{\left({C}_{pred,i}-{C}_{obs,i}\right)}^{2}}$$

where C_pred, i and C_obs, i represent predicted and observed concentrations respectively, and n is the number of data points42.

Table 3 Comparative performance evaluation of different modeling approaches for microplastic transport prediction.
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The comparative analysis reveals that the proposed spatiotemporal graph neural network consistently outperforms traditional numerical models, machine learning approaches, and conventional deep learning methods across multiple evaluation metrics. Figure 4 illustrates the performance comparison across varying flow conditions and temporal horizons. The Nash-Sutcliffe efficiency coefficient, calculated as:

$$\:NSE = 1 - \frac{{\sum\nolimits_{{i = 1}}^{n} {\left( {C_{{obs,i}} - C_{{pred,i}} } \right)^{2} } }}{{\sum\nolimits_{{i = 1}}^{n} {\left( {C_{{obs,i}} - \overline{C} _{{obs}} } \right)^{2} } }}$$

where \(\:\overline{C} _{{obs}}\) represents the mean of observed values, achieved a value of 0.892, indicating excellent model performance for hydrological prediction applications43.

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Model performance analysis across flow conditions and spatial scales. (a) Prediction accuracy under varying flow regimes showing R² = 0.923 for low-flow and R² = 0.887 for high-flow conditions. (b) Spatial accuracy distribution demonstrating effective upstream-downstream relationship capture with correlation coefficients > 0.85 across all reaches.

The model generalizability validation was conducted across three distinct river system types to assess cross-domain applicability. In mountainous steep rivers (slope > 0.01), the model achieved R² = 0.834, effectively capturing rapid transport dynamics despite high velocity gradients. For meandering lowland rivers (sinuosity > 1.5), performance reached R² = 0.876, successfully modeling complex recirculation patterns and particle trapping mechanisms. Urban channelized systems showed R² = 0.851, demonstrating robust adaptation to artificial hydraulic structures and modified flow regimes. The transfer learning approach required only 15% of local calibration data to achieve > 90% of full-training performance, confirming strong generalizability across diverse geographical and climatic conditions.

The model applicability assessment across different river conditions demonstrates robust performance across diverse hydrodynamic regimes, from low-flow periods characterized by particle settling dominance to high-flow events where advective transport prevails.

The stability analysis reveals consistent prediction accuracy with coefficient of variation less than 15% across different seasonal conditions and flow magnitudes, indicating reliable performance for operational applications. The model maintains high accuracy even during extreme flow events, with only marginal performance degradation compared to baseline conditions.

The prediction error analysis reveals that the largest uncertainties occur during rapid flow transitions and at tributary confluences where complex three-dimensional flow patterns challenge the graph-based representation. The error distribution follows approximately normal patterns with mean absolute error remaining below 5% for most prediction scenarios. Spatial error analysis indicates higher prediction uncertainties in reach transitions and hydraulic control sections, while temporal errors primarily occur during storm events when rapid changes in flow conditions exceed the model’s adaptation capacity. The systematic error analysis demonstrates minimal bias in concentration predictions, with residual patterns showing no significant correlation with environmental variables, confirming the model’s unbiased estimation capability across different transport conditions.

Analysis of hydrodynamic factor influence mechanisms

The analysis of flow velocity effects on microplastic transport pathways reveals a strong nonlinear relationship where particle trajectory deviations increase exponentially with velocity magnitude44. Table 4 presents the sensitivity analysis of hydrodynamic factors affecting microplastic transport mechanisms. The transport distance coefficient, defined as the ratio of actual transport distance to theoretical advective distance, demonstrates significant variation across different velocity regimes. Under low-velocity conditions (< 0.3 m/s), microplastics exhibit enhanced settling behavior with transport coefficients ranging from 0.4 to 0.6, while high-velocity flows (> 1.0 m/s) maintain particles in suspension with transport coefficients approaching unity.

Table 4 Sensitivity analysis of hydrodynamic factors affecting microplastic transport mechanisms.
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The influence of water depth on microplastic deposition patterns follows a power-law relationship that can be expressed as:

$$\:{D}_{eff}={D}_{0}\cdot\:{\left(\frac{h}{{h}_{0}}\right)}^{-\alpha\:}$$

where D_eff is the effective deposition rate, D_0 is the reference deposition rate, h is the local water depth, h_0 is the reference depth, and α is the depth-dependency exponent with values typically ranging from 0.6 to 1.2 depending on particle characteristics45.

Channel curvature analysis demonstrates that meandering sections create distinct microplastic accumulation zones through secondary circulation patterns that enhance lateral transport and promote particle trapping in inner bank regions. The curvature influence coefficient, calculated as the ratio of lateral to longitudinal transport rates, shows maximum values at bend apexes where centrifugal forces reach peak intensity. The aggregation mechanism in curved channels follows a helical flow pattern that concentrates buoyant microplastics toward the outer bank surface while promoting deposition of denser particles in inner bank shallow zones.

The quantification of factor interactions reveals significant synergistic effects between flow velocity and turbulence intensity, with their combined influence exceeding the sum of individual contributions by approximately 25%. Figure 5 provides a quantitative analysis of hydrodynamic factor influence weights and interaction effects on microplastic transport. The interaction parameter can be expressed as:

$$\:{I}_{ij}=\frac{{\partial\:}^{2}C}{\partial\:{F}_{i}\partial\:{F}_{j}}\cdot\:{\sigma\:}_{i}{\sigma\:}_{j}$$

where I_ij represents the interaction strength between factors i and j, C is the microplastic concentration response, F_i and F_j are the respective factor values, and σ_i and σ_j are the standard deviations of the factors46. The analysis identifies flow velocity as the primary controlling factor with a relative importance of 34.2%, followed by bed shear stress at 28.7% and turbulence intensity at 21.1%. The dispersal mechanisms under varying hydrodynamic conditions demonstrate that high-energy environments promote uniform particle distribution through enhanced mixing, while low-energy conditions facilitate size-selective sorting and preferential accumulation in depositional areas such as channel margins and backwater zones.

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Hydrodynamic factor influence analysis. (a) Factor importance ranking showing flow velocity (34.2%), bed shear stress (28.7%), and turbulence intensity (21.1%) as dominant controls. (b) Key factor interaction effects revealing 25% synergistic enhancement between velocity and turbulence intensity.

Spatiotemporal scale effects and prediction accuracy analysis

The investigation of model predictive capability across different temporal and spatial scales reveals distinct performance patterns that reflect the underlying physics of microplastic transport processes47. Table 5 summarizes the multi-scale prediction accuracy statistics for microplastic transport modeling. Short-term predictions (1–6 h) demonstrate exceptional accuracy with correlation coefficients exceeding 0.92, primarily due to the dominance of advective transport mechanisms that are well-captured by the spatiotemporal graph neural network architecture. The prediction accuracy gradually decreases with extended forecast horizons, with medium-term predictions (1–7 days) showing correlation coefficients around 0.85, while long-term predictions (1–4 weeks) maintain acceptable performance levels with correlations above 0.72.

Table 5 Multi-scale prediction accuracy statistics for microplastic transport modeling.
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The spatial scale analysis demonstrates that the model maintains high accuracy for reach-scale predictions (100 m-1 km) with minimal performance degradation compared to point-scale estimations, indicating effective capture of local transport mechanisms and spatial correlation patterns. However, watershed-scale predictions (> 10 km) show increased uncertainty due to cumulative effects of model approximations and the influence of unmonitored tributary inputs and boundary conditions. The model performance exhibits optimal accuracy at intermediate spatial scales (1–5 km) where the graph structure effectively represents the dominant flow connectivity while avoiding excessive computational complexity associated with fine-scale discretization.

The accuracy differential analysis between short-term and long-term predictions reveals that the primary source of performance degradation stems from the accumulation of meteorological uncertainties and the influence of external forcing factors not explicitly included in the hydrodynamic input variables. Short-term predictions benefit from the persistence of current hydrodynamic conditions and the relative stability of particle transport patterns, while long-term forecasts encounter challenges from seasonal variations in flow regimes, temperature-dependent settling velocities, and biofilm formation effects on particle properties. The temporal error propagation analysis indicates that prediction uncertainty increases approximately logarithmically with forecast horizon, following patterns consistent with chaotic dynamical systems where small initial errors amplify over time.

The multi-scale transport process evaluation demonstrates that the spatiotemporal graph neural network effectively captures transport phenomena operating at different scales simultaneously, from turbulent mixing processes at sub-meter scales to seasonal migration patterns at basin scales48. The model shows particular strength in representing intermediate-scale processes such as meander-induced transport and tributary confluence mixing, which are critical for understanding reach-scale microplastic distribution patterns. The scale effect implications for practical applications suggest that the model is most suitable for operational forecasting at time horizons of 1–7 days and spatial extents of 1–10 km, providing sufficient accuracy for environmental management decisions while maintaining computational efficiency. For longer-term strategic planning applications, the model can provide valuable trend analysis and scenario assessment capabilities, though with appropriately adjusted confidence intervals to account for increased prediction uncertainty at extended temporal scales.

Conclusions

This research presents a novel spatiotemporal graph neural network framework for elucidating the influence mechanisms of river hydrodynamics on microplastic transport processes, providing significant scientific contributions to environmental modeling and pollution assessment methodologies. The primary innovation lies in the integration of graph-based river network representation with multi-scale temporal feature extraction, enabling comprehensive capture of complex spatiotemporal dependencies that govern microplastic migration patterns in riverine environments. The developed model successfully incorporates physics-informed constraints within the neural network architecture, ensuring that predictions remain consistent with fundamental transport principles while leveraging the flexibility of machine learning approaches49.

The model demonstrates superior predictive performance compared to conventional numerical models and traditional machine learning methods, achieving correlation coefficients exceeding 0.89 for microplastic concentration predictions across diverse hydrodynamic conditions. The computational efficiency advantages are particularly notable, with processing times reduced by approximately 92% compared to traditional numerical approaches while maintaining comparable or superior accuracy levels. The adaptive attention mechanism successfully identifies critical hydrodynamic factors, revealing that flow velocity and bed shear stress constitute the dominant controls on microplastic transport, accounting for 62.9% of the total variance in concentration patterns.

Despite these achievements, several limitations warrant acknowledgment and future investigation. The current model relies on simplified representations of particle-particle interactions and biofilm formation effects, which may influence long-term transport behavior under certain environmental conditions. The graph structure representation, while effective for main channel transport, may inadequately capture complex three-dimensional flow patterns in highly irregular channel geometries or during extreme flood events. Additionally, the model’s performance shows degradation for predictions beyond weekly time horizons due to accumulated uncertainties in meteorological forcing and boundary conditions.

The application prospects for this methodology in river environmental management and microplastic pollution control appear highly promising, particularly for real-time monitoring system design, pollution source identification, and remediation strategy optimization50. The framework provides valuable capabilities for scenario analysis and impact assessment of different management interventions, supporting evidence-based decision-making for pollution control policies. The model’s ability to quantify uncertainty and provide confidence intervals enhances its utility for risk assessment applications where decision reliability is critical.

Future research directions should focus on incorporating particle-sediment interaction mechanisms, developing multi-species transport capabilities for different microplastic types, and extending the framework to handle complex river networks with multiple tributaries and hydraulic structures. The integration of remote sensing data and continuous monitoring systems could further enhance model calibration and validation capabilities. Additionally, the development of transfer learning approaches would enable model adaptation across different river systems without extensive retraining, facilitating broader application of the methodology in diverse geographic and climatic contexts.