Geometry-Aware Latent Representation Learning for Drag Coefficient Prediction of Irregular Particles

1. Introduction

• For irregular particles, geometrically distinct shapes can share hand-crafted descriptor values while producing different drag responses.Descriptors summarize only selected aspects of a three-dimensional shape.Source: §1 · p. 2
• The predictive value of learned versus hand-crafted geometric representations remains unclear as geometric complexity increases and test geometries depart from training variations.Earlier learned-representation studies did not systematically isolate predictive value against conventional descriptors under controlled changes in complexity and training coverage.Source: §1 · pp. 2–3
• VAE-MLP jointly learns a compact geometry-aware representation through geometry reconstruction, drag prediction, and latent regularization.The framework is compared with a descriptor-based MLP under matched evaluation settings.Source: §1 · p. 3
• The study tests geometric complexity and training coverage using regular ellipsoids, overall-shape and fine-scale surface variations, and evaluation-only fine-scale variations.The experiments separately assess changes represented during training and fine-scale variations introduced only at evaluation.Source: §1 · p. 3
• VAE-MLP's advantage is most pronounced for unseen fine-scale surface variations and persists across five training seeds.The models also differ in how they use geometric information to predict drag on those unseen geometries.Source: §1 · p. 3

2.1. Problem Formulation

• The prediction formulation uses Reynolds number and a geometric representation of the particle; the representation is either learned or hand-crafted.Descriptor-MLP uses prescribed descriptors, whereas VAE-MLP learns from voxelized three-dimensional geometry.Source: §2.1 · p. 4

2.2. Geometry-Aware Latent Representation Learning with VAE-MLP

• VAE-MLP voxelizes particle geometry and jointly trains an encoder, decoder, and drag regressor so the latent representation supports reconstruction and drag prediction.The drag regressor combines the latent vector with Reynolds number; KL regularization constrains the latent distribution.Source: §2.2 · p. 4, Figure 1 · p. 5

2.3. Variational Latent Encoding

• The variational encoder maps voxelized geometry to a posterior distribution, and the reparameterization trick enables sampling during training.The regressor uses sampled latent vectors during training and the posterior mean for deterministic inference.Source: §2.3 · p. 5

2.4. Reconstruction and Drag-Prediction Branches

• The decoder reconstructs voxel occupancy from the latent vector, while the drag regressor combines that vector with Reynolds number to predict drag.At inference, the posterior mean replaces the sampled latent vector.Source: §2.4 · p. 6

2.5. Loss Function: Joint Training Objective

• The encoder, decoder, and regressor are jointly optimized with reconstruction, KL-regularization, and drag-prediction losses.The reconstruction objective uses positive-class-weighted binary cross-entropy for sparse voxel occupancy; the drag loss is a target-magnitude-weighted squared error.Source: §2.5 · pp. 6–7

3.1.1. Particle geometry generation

• Ellipsoids span aspect ratios Ar ∈[0.25, 2.5], while spherical-harmonic perturbations use dL for degrees 2–8 and dH for degrees 9–14.The perturbation coefficients are independently sampled for each geometry; perturbation is applied without an additional volume-rescaling step, and geometries are rotated to impose incident angles αinc ∈[0°, 90°].Source: §3.1.1 · pp. 7–8

3.1.2. Voxel occupancy fields

• Oriented particle meshes are converted to binary occupancy fields and centered in fixed 64 × 64 × 64 arrays for the three-dimensional convolutional network.The mesh is translated and scaled so its longest bounding-box dimension is 0.9, then surface voxels are generated, interiors filled, and the array zero-padded.Source: §3.1.2 · pp. 8–9

3.2. Network architecture and training

• VAE-MLP combines a three-dimensional convolutional encoder and decoder with an MLP drag regressor; main experiments use latent dimension d = 16, except E1, which uses d = 2.The loss weights are fixed across experiments, and models are trained for 200 epochs with Adam; the checkpoint with minimum validation drag loss is retained.Source: §3.2 · p. 9

3.3. CFD reference datasets

• Drag targets come from transient incompressible-flow simulations over Re ∈[10, 200], with the final target obtained by averaging the latter portion of each drag-coefficient history.The reference area is based on the volume-equivalent sphere, and force is evaluated from pressure and skin-friction contributions over the particle surface.Source: §3.3 · p. 9

3.4. Dataset partition and experimental regimes

• Training and validation are split by intrinsic geometry to prevent geometry-level leakage, and prediction metrics are evaluated on independently generated external datasets.E2b and E3 share an 864-case external grid; aggregate evaluation uses all 864 E2b cases and 648 E3 cases with dH > 0, reserving 216 dH = 0 cases as matched references.Source: §3.4 · p. 10, Table 1 · p. 10

3.5. Comparison baselines

• Descriptor-MLP tests hand-crafted geometric inputs against learned representations, while empirical correlations provide established references for regular ellipsoids and irregular particles.Descriptor-MLP uses Deq, sphericity Φ, crosswise sphericity Φcross, and Reynolds number; its regression backbone and training settings match the VAE-MLP drag-prediction branch.Source: §3.5 · pp. 10–11

3.6. Evaluation metrics

• Mean absolute relative error (MARE) is the primary aggregate metric, with maximum relative error, condition-wise MARE, and R2 providing complementary views of prediction performance.Condition-wise MARE is computed over samples at fixed (Re, dL, dH), and cross-model comparisons use identical evaluation samples and configurations.Source: §3.6 · pp. 11–12

3.7. Experimental matrix

• E1 evaluates regular ellipsoids, E2a low-order irregular particles, E2b low- and high-order perturbations, and E3 high-order perturbations excluded from training but present in evaluation.E2b includes dH ∈[0, 0.3], whereas E3 trains with dH = 0 and evaluates particles with dH > 0.Source: §3.4 · p. 10, §3.7 · p. 12, Table 2 · p. 12

4.1. E1: Learned latent representation and drag coefficient prediction for regular ellipsoids

• The two-dimensional E1 latent space varies continuously with aspect ratio and incident angle, although neither is supplied as an explicit scalar input.Near-spherical geometries cluster centrally, while oblate and prolate geometries extend along distinct trajectories.Source: §4.1 · p. 13, Figure 4 · p. 13
• On regular ellipsoids, VAE-MLP and Descriptor-MLP have similar MAREs, 2.80% and 2.77%, while the Ke correlation [7] has 5.86%.Maximum absolute relative errors are 12.97% for VAE-MLP, 30.37% for Descriptor-MLP, and 11.37% for the Ke correlation [7].Source: §4.1 · pp. 13–14, Table 3 · p. 14, Figure 5 · p. 14

4.2. E2a and E2b: Interpolative study under geometric perturbations

• On represented irregular geometries, VAE-MLP has lower MARE than Descriptor-MLP in E2a (3.37% versus 4.39%) and E2b (2.85% versus 3.59%).In E2b, both models have R2 = 0.994; VAE-MLP also has a lower maximum absolute relative error, 17.85% versus 26.60%.Source: §4.2 · pp. 14, 16, Table 4 · p. 15, Table 5 · p. 17

4.3.1. Predictive performance

• Across five runs, unseen-perturbation MARE is 3.99% ± 0.55% for VAE-MLP and 6.11% ± 1.72% for Descriptor-MLP; at dH = 0.3, it is 4.97% ± 0.78% and 9.88% ± 3.22%.VAE-MLP has lower aggregate MARE in all five runs with a fixed data split.Source: Table 6 · p. 18, §4.3.1 · p. 18

4.3.2. Analysis of extrapolative performance

• At dH = 0.3, Descriptor-MLP overpredicts the mean CFD drag increment, whereas VAE-MLP underpredicts it; perturbed-geometry latent inputs lower VAE-MLP MARE in all five runs.For VAE-MLP, MARE is 4.97% using the perturbed geometry's latent input and 6.34% using the matched reference input.Source: §4.3.2 · p. 22
• The two models differ in their response to unseen perturbations: Descriptor-MLP has higher correlation and sign agreement with CFD differences, while VAE-MLP has lower increment MAE.Across five runs, correlation is 0.733 versus 0.577, sign agreement is 90.59% versus 64.91%, and increment MAE is 0.0876 versus 0.1025 for Descriptor-MLP and VAE-MLP, respectively.Source: §4.3.2 · p. 20, Table 8 · p. 22

5. Conclusions

• The jointly trained VAE-MLP reconstructs voxelized geometry and predicts drag using the latent representation and Reynolds number.The encoder, decoder, and regressor are optimized with reconstruction, KL-regularization, and drag-prediction losses.Source: §5 · p. 22
• For represented variations, the models have comparable average accuracy on regular ellipsoids, while VAE-MLP has lower errors on irregular geometries.The irregular cases span overall-shape and fine-scale surface variations.Source: §5 · p. 24
• When high-order surface variations are absent from training, VAE-MLP has lower overall MARE than Descriptor-MLP in all five runs: 3.99% ± 0.55% versus 6.11% ± 1.72%.At dH = 0.3, mean MAREs are 4.97% for VAE-MLP and 9.88% for Descriptor-MLP.Source: §5 · p. 24
• At dH = 0.3, VAE-MLP's perturbed-geometry latent inputs improve MARE in all five runs, while Descriptor-MLP overestimates mean CFD drag increment.Descriptor-MLP's elevated error is associated with its fitted response to changes in sphericity and crosswise sphericity.Source: §5 · p. 24
• At fixed spatial resolution, binary voxels have limited ability to resolve fine-scale surface features; the study focuses on drag coefficient prediction.The authors identify signed distance functions and other hydrodynamic responses, including lift and torque, as future work.Source: §5 · p. 24

CRediT authorship contribution statement

• Li led conceptualization, formal analysis, data curation, methodology, software, validation, visualization, and drafting; other authors contributed to methods, review, supervision, or project resources.The listed roles also include funding acquisition and project administration.Source: CRediT authorship contribution statement · p. 24

Appendix A. Spherical-harmonic geometry construction

• Particle surfaces use separate spherical-harmonic expansions for the Cartesian coordinates, truncated at N = 15, with real parts retained for surface vertices.Low-order degrees are n = 2–8 and high-order degrees are n = 9–14; dL and dH control those perturbation bands.Source: Appendix A · p. 25
• Each particle receives an independently generated coefficient realization, so different dL and dH levels are not nested rescalings of the same perturbation.The mapped surface mesh has 5120 triangular facets, and the prescribed incident angle is imposed after perturbation.Source: Appendix A · p. 26

Appendix B. CFD formulation and reference-data generation

• The CFD simulations model a rigid particle held at a prescribed position and orientation in incompressible Newtonian flow.The described setup is implemented in OpenFOAM v2306.Source: Appendix B · p. 26

Appendix C. Detailed dataset and evaluation configurations

• Training flow conditions are sampled over incident angles from 0° to 90° and Reynolds numbers from 10 to 200 using two-dimensional Latin hypercube sampling.Training geometries are sampled within the particle parameter domains specified in Table C.10.Source: Appendix C · p. 29, Table C.10 · p. 29
• All orientations and flow conditions from one intrinsic geometry are assigned together to either training or validation data.The partition is performed at the intrinsic-geometry level.Source: Appendix C · p. 29
• The E1 latent-space visualization uses 150 voxelized geometries from a grid of 15 aspect ratios and 10 incident angles, without CFD drag targets.Aspect ratios span 0.5–2.0 logarithmically, and incident angles span 0°–90° uniformly.Source: Appendix C · p. 30

Appendix D.1. Architecture specification

• The VAE-MLP uses the same layer configuration across the main experiments, with latent dimension d = 2 for E1 and d = 16 for E2a, E2b, and E3.The drag regressor takes the latent representation concatenated with Reynolds number, giving input dimension d + 1.Source: Appendix D.1 · p. 30

Appendix D.2. Empirical drag correlations

• The appendix evaluates the ellipsoid-specific Ke et al. and crosswise-sphericity Hölzer–Sommerfeld empirical drag correlations.Source: Appendix D.2 · p. 31

Appendix D.3. implementation and random seed detail

• E1, E2a, and E2b use seed 42, while E3 uses five runs with seeds 42–46 on the same dataset partition.Detailed E3 figures show seed 46; input-substitution tables summarize all five runs.Source: Appendix D.3 · p. 31

• The comparisons match each perturbed condition to regular ellipsoids with the same aspect ratio, incident angle, and Reynolds number.The evaluated parameter combinations include Ar, αinc, and Re, with independently generated perturbed realizations.Source: Appendix E · p. 32
• At dH = 0.3, mean drag increases are 9.14% and 13.88% for dL = 0.25 and 0.35, versus 2.57% and 5.03% at dH = 0.Figure E.16 reports mean signed relative drag change averaged over 84 matched samples per perturbation condition.Source: Appendix E · p. 32, Figure E.16 · p. 32

• 5.73% is the maximum VAE-MLP cMARE at Re = 200, dL = 0.35, and dH = 0.3; Descriptor-MLP reaches 7.28% at Re = 75 with the same perturbation levels.Figure F.17 groups E2b MARE by dL, while Figure F.18 maps it over (dL, dH) at selected Reynolds numbers.Source: Appendix F · p. 32

• Mean IoU rises from approximately 0.57 at d = 3 to 0.66 at d = 32 across 864 external evaluation samples, with smaller improvement beyond d = 16.Reconstruction continues to improve over the evaluated range.Source: Appendix G · p. 34
• Overall MARE is lowest at d = 16, although the dimension with lowest level-specific MARE varies with dH.The minima occur at d = 12 for dH = 0, d = 16 for dH = 0.1, and d = 24 for dH = 0.2 and 0.3.Source: Appendix G · p. 34
• The study uses d = 16 for the main E2a, E2b, and E3 experiments, but does not establish it as optimal for other geometry families or training regimes.Source: Appendix G · p. 34

Input-substitution design

• Reference-based substitution changes the geometry input while keeping the trained predictor and Reynolds number fixed, and evaluates both inputs against the same perturbed-geometry CFD target.Matched geometries share aspect ratio, incident angle, and dL; they are independent realizations under those conditions.Source: Appendix H · pp. 34–35

Evaluation scope

• The reported strongest-perturbation diagnostics use dH = 0.3, 24 matched geometry pairs, nine Reynolds numbers, and five training seeds.MARE is computed over 216 flow observations per seed; the data split is fixed at seed 42.Source: Appendix H · p. 35

Descriptor restoration

• Restoring either area-related descriptor substantially reduces Descriptor-MLP error, whereas restoring Deq has little effect, consistently across five runs.The results associate elevated error with the model's response to changes in Φ and Φcross.Source: Appendix H · p. 35, Table H.13 · p. 36

Latent-input substitution

• At dH = 0.3, using the perturbed geometry’s latent representation lowers cross-seed mean MARE from 6.34% to 4.97% versus the matched reference representation, with improvement in all five training runs.Reductions range from 1.01 to 1.73 percentage points.Source: Appendix H · p. 36, Table H.14 · p. 36