Generative modeling of granular flow on inclined planes using conditional flow matching

Xuyang Li; Rui Li; Teng Man; Yimin Lu

cs.CEApr 6, 2026

Generative modeling of granular flow on inclined planes using conditional flow matching

Authors: Xuyang Li, Rui Li, Teng Man, Yimin Lu

Organizations: School of Construction, University of North Carolina at Charlotte, Charlotte, NC 28223, USA · Department of Civil, Environmental, and Construction Engineering, Texas Tech University, Lubbock, TX 79409, USA · College of Civil Engineering, Zhejiang University of Technology, Hangzhou, Zhejiang 310023, China

Abstract

Granular flows govern many natural and industrial processes, yet their interior kinematics and mechanics remain largely unobservable, as experiments access only boundaries or free surfaces. Conventional numerical simulations are computationally expensive for fast inverse reconstruction, and deterministic models tend to collapse to over-smoothed mean predictions in ill-posed settings. This study, to the best of the authors' knowledge, presents the first conditional flow matching (CFM) framework for granular-flow reconstruction from sparse boundary observations. Trained on high-fidelity particle-resolved discrete element simulations, the generative model is guided at inference by a differentiable forward operator and a novel sparsity-aware gradient guidance mechanism. This mechanism avoids the gradient dilution inherent to standard mean-squared-error approaches, preserves the absolute physical scale of observation errors, enforces measurement consistency without hyperparameter tuning, and prevents unphysical velocity predictions in non-material regions. A physics decoder maps the reconstructed velocity fields to stress states and energy fluctuation quantities, including mean stress, deviatoric stress, and granular temperature. The framework accurately recovers interior flow fields from full observation to only 16% of the informative window, and it remains effective under strongly diluted spatial resolution with only 11% of data. It also outperforms a deterministic CNN baseline in the most ill-posed reconstruction regime and provides spatially resolved uncertainty estimates through ensemble generation. These results demonstrate that conditional generative modeling offers a practical route for non-invasive inference of hidden bulk mechanics in granular media, and it suggests potential applicability for inverse problems in particulate and multiphase systems.

Figures & tables

Figure 1: Numerical modeling and dataset construction. (a) Schematic view of the discrete element method (DEM) model showing the geometry, particle size, coordinate system, and representative surfaces of interest, from the observational boundary (Slice 0) to the other boundary plane (Slice 3). (b) Representative DEM results, showing runout velocity

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profile after packing, during the flow, and after flow stops. (c) Demonstration of the grid-averaging scheme that transfers particle-scale information into grid cells , with a grid size of 4 mm. (d) Demonstrative grid-averaged results at

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s used for conditional flow matching, including runout velocity

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, granular temperature

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, mean stress

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, and deviatoric stress

\mathchar 29041

Figure 2: Overview of the proposed framework. The architecture integrates (a) a continuous-time probabilistic flow matching backbone as a kinematic prior $\mathchar 29030$ , (b) a differentiable neural forward operator $\mathchar 29031$ with a gradient guidance mechanism for conditional flow matching to infer internal particle dynamics from partial boundary observations, and (c) a physics decoder $\mathcal{\mathchar 29000}$ to infer key physical properties from the reconstructed kinematic fields.

Figure 3: Validation of the generative kinematic prior. (a) Empirical cumulative distribution functions (eCDFs) of the three velocity components ( $\delimiter 69640972_{\mathchar 29048}$ , $\delimiter 69640972_{\mathchar 29049}$ , and $\delimiter 69640972_{\mathchar 29050}$ ). (b) Cumulative probability distributions of spatial velocity gradients, including shear strain rates ( $\dot{\mathchar 28941}_{\mathchar 29048}\mathchar 24891\dot{\mathchar 28941}_{\mathchar 29050}$ ) and volumetric divergence ( $\mathchar 626\mathchar 8705\mathbf{\delimiter 69640972}$ ). The unguided generative samples show a strong overall match with the DEM ground truth, demonstrating that the model successfully captures both the underlying kinematic distribution and spatial-physical constraints.

Figure 4: Reconstruction of internal velocity fields from full boundary observations at $\mathchar 29044\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28725\mathchar 28726$ s. The top and bottom rows display the streamwise ( $\delimiter 69640972_{\mathchar 29048}$ ) and vertical ( $\delimiter 69640972_{\mathchar 29050}$ ) velocity components, respectively. From left to right, the columns present the boundary observation (Slice 0) used as the conditioning input, the DEM ground truth at the deepest internal plane (Slice 3), the corresponding CFM reconstruction, and the absolute error field.

Figure 5: Reconstruction of internal streamwise velocity fields ( $\delimiter 69640972_{\mathchar 29048}$ ) from partial boundary observations at $\mathchar 29044\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28725\mathchar 28726$ s. (a) The masked input boundary observation region, indicated by a pink bounding box. (b) Visualization of the observation window on the boundary (Slice 0) and prediction Slices 1-3. (c) Slice-by-slice comparison across three internal depths (rows: Slices 1, 2, and 3 from top to bottom). The columns display the DEM ground truth with green boxes showing the projection of the boundary observation window, the CFM reconstruction, and the absolute error, respectively. (d) Vertical profiles of velocity ( $\delimiter 69640972_{\mathchar 29048}$ ) and shear strain rate ( $\dot{\mathchar 28941}_{\mathchar 29048}$ ) at $\mathchar 29048\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28720$ m and $\mathchar 29048\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28721$ m across two flow stages ( $\mathchar 29044\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28723\mathchar 28726$ and $\mathchar 28720\mathchar 314\mathchar 28727\mathchar 28726$ s) along $\mathchar 29050$ direction. The plots compare the ground truth and the CFM prediction at two different internal depths (Slice 1 and Slice 3).

Figure 6: End-to-end inference of physical fields from partial boundary observations at $\mathchar 29044\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28725\mathchar 28726$ s for Slice 3 . (a) Spatial distributions of pressure $\mathchar 29040$ , shear stress $\mathchar 29041$ , and granular temperature $\mathchar 29012$ at the deepest unobservable plane. The columns display the DEM ground truth, the chained CFM prediction, and the absolute spatial error, respectively. (b) Depth-wise profiles of $\mathchar 29040$ , $\mathchar 29041$ , and the effective friction coefficient ( $\mathchar 28950\mathchar 12349\mathchar 29041\delimiter 68408078\mathchar 29040$ ) at $\mathchar 29048\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28720$ m, comparing the raw data and linear fits between the ground truth and CFM predictions.

Figure 7: Comparison of the reconstructed streamwise velocity ( $\delimiter 69640972_{\mathchar 29048}$ ) between the generative CFM framework and the deterministic CNN baseline at the deepest interior plane (Slice 3) for $\mathchar 29044\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28725\mathchar 28726$ s. The columns display the DEM ground truth, CFM and CNN predictions, and their respective absolute error maps. (a) Results under full boundary observation. (b) Results under severe partial observation.

Figure 8: Uncertainty analysis of the internal flow reconstruction under partial observation at

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s. (a) Slice-by-slice spatial evaluation across three internal depths, displaying the CFM ensemble mean, absolute error, and pixel-wise standard deviation. (b) 1D velocity propagation profile along the thickness (

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) direction extracted at a sample boundary location (marked by the cross ‘

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’). The plot compares the depth-wise ground truth velocity (black dashed line) against the CFM ensemble mean (red line) and the full distribution spread of all 25 raw generative ensembles (gray lines).

Figure 9: Ablation study comparing the proposed sparsity-aware guidance (top row) and two baselines: normalized guidance (middle row) and MSE guidance (bottom row) . The columns display the reconstructed velocity field, a zoom-in view of the empty region, and the global absolute error map without background masking. The normalized baseline introduces severe non-physical numerical noise in empty and zero-velocity pixels and corrupts the active flow profile, while the MSE baseline suffers from severe gradient dilution . In contrast , the proposed method accurately maintains sharp material boundaries and absolute zero values in those spaces.

Figure 10: Sensitivity analysis of the reconstruction fidelity under varying observation constraints (evaluated on Slice 3 at $\mathchar 29044\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28721\mathchar 28726\mathchar 24891\mathchar 28720\mathchar 314\mathchar 28725\mathchar 28726\mathchar 24891$ and $\mathchar 28720\mathchar 314\mathchar 28727\mathchar 28726$ s ). Left: Robustness against restricted spatial coverage, where the horizontal axis represents the varying coverage ratio $\mathchar 28954$ . Representative input conditions at $\mathchar 28954\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28728$ and $\mathchar 28954\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28724$ are visualized as insets. Right: Robustness against diluted spatial resolution, where the horizontal axis denotes the spatial subsampling stride $\mathchar 29035$ , with insets illustrating the sparse inputs at $\mathchar 29035\mathchar 12349\mathchar 28723$ and $\mathchar 29035\mathchar 12349\mathchar 28727$ . In both plots, solid lines with varying color shades represent the active RMSE at different flow stages , and the corresponding dashed lines represent the spatial Pearson correlation $\mathchar 29042$ (right vertical axis).

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 11: Velocity field reconstruction under partial boundary observations at additional time steps. (a) Reconstructed slices at $\mathchar 29044\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28721\mathchar 28726$ s. (b) Reconstructed slices at $\mathchar 29044\mathchar 12349\mathchar 28720\mathchar 314\mathchar 28723\mathchar 28726$ s. Both panels compare the DEM ground truth, the generative prediction, and the absolute spatial error across three distinct internal depths.

Figure 12: Sensitivity analysis of the reconstruction fidelity evaluated on Slice 2 across three different macroscopic flow stages (

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and

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s). The relative degradation patterns remain consistent with the deepest slice.

Figure 13: Sensitivity analysis of the reconstruction fidelity evaluated on Slice 1 across three different macroscopic flow stages (

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and

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s). This slice is closest to the observation boundary, yielding the lowest absolute reconstruction errors while maintaining the same robustness threshold.

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PerFlow: Physics-Embedded Rectified Flow for Efficient Reconstruction and Uncertainty Quantification of Spatiotemporal Dynamics

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Reconstructing PDE-governed fields from sparse and irregular measurements is challenging due to their ill-posed nature. Deterministic surrogates are trained on dense fields that struggle with limited measurements and uncertainty quantification. Generative models, by learning distributions over spatiotemporal fields, can better handle sparsity and uncertainty. However, existing generative approaches enforce data consistency and PDE constraints simultaneously via sampling-time gradient guidance, resulting in slow and unstable inference. To this end, we propose PerFlow, a Physics-embedded rectified Flow for efficient sparse reconstruction and uncertainty quantification of spatiotemporal dynamics. PerFlow decouples observation conditioning from physics enforcement, performing guidance-free conditioning by feeding observations into rectified-flow dynamics while embedding hard physics via a constraint-preserving projection (e.g., incompressibility or conservation). Theoretically, we establish invariance guarantees to ensure that trajectories remain on the physics-consistent manifold throughout sampling. Experiments on various PDE systems demonstrate competitive reconstruction accuracy with sound physics consistency, while enabling efficient conditional sampling (e.g., 50 steps) and up to 320x faster inference than 2000-step guided diffusion baselines.

Abstract

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