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A shape-similarity latent space for fluid interfaces: invertible reduced-order modelling of droplet morphology

Ali R. Hashemi, Mohammad R. Hashemi, Pavel B. Ryzhakov

arXiv:2609.37947v1physics.flu-dynphysics.comp-phphysics.data-anstat.ML

TL;DR

Sparse recordings and costly simulations leave intermediate droplet interfaces unavailable, although similar morphologies may occur elsewhere in a campaign. SHROM combines an elastic shape-space graph with an autoencoder, and the graph term establishes a correspondence between latent position and morphology without improving reconstruction.

  • Problem

    Sparse high-speed recordings and costly simulations make intermediate droplet states difficult to obtain, while elastic shape analysis alone supplies no reduced coordinates.

    Source: §1 · p. 2, §2.1 · p. 4

  • Method

    SHROM represents interfaces with square-root velocity functions, builds a neighbour graph over that shape space, and trains an autoencoder to reconstruct while preserving graph neighbourhoods in its latent.

    Source: Approach · p. 2

  • Results

    Removing the graph term leaves reconstruction unchanged but destroys the correspondence between latent position and shape morphology.

    Source: §7 · p. 32

  • Takeaways & Limitations

    A state missing from one sequence can be recovered from the corpus by decoding the latent position corresponding to its shape likeness.

    Source: §7 · p. 32

    Interpolation across a topology change is a representational failure: the latent region traversed contains no valid intermediate shapes, although position corresponds to morphology elsewhere.

    Source: §6.2 · pp. 29–30

Abstract

from arXiv · show

A droplet breaks up in tens of microseconds, and a recording captures perhaps a dozen frames. The states in between cannot be recovered without repeating the experiment, and simulating them is too costly to sweep an operating envelope. Yet they are present in the corpus as a whole: a campaign spanning a device's actuation range produces morphologies resembling those any single recording missed. Exploiting that requires a representation that is low-dimensional, invertible, and faithful to shape rather than to sampling. Proper orthogonal decomposition supplies the first two but measures distance in sampled coordinates; manifold learning supplies the third but no map back to a shape; elastic shape analysis supplies a shape metric but no reduced coordinates. SHROM composes all three. Interfaces are represented by their square-root velocity functions, a neighbour graph is built over that shape space, and an autoencoder is trained to reconstruct while penalising latents in which graph neighbours are not latent neighbours. The demonstration uses 301,539 inkjet droplet contours and four filmed break-up sequences. The graph term does not improve reconstruction. It determines whether position in the latent carries meaning: clustering the latent of an otherwise identical model recovers the shape-space partition at chance level (ARI = 0.063 +/- 0.059), and at 0.781 +/- 0.049 with the term active. Waveform parameters predict the full contour at R^2 = 0.878 +/- 0.026. Negative results are reported in the same terms. The graph metric proved immaterial across five choices, and interpolation error saturates at the reconstruction limit, so a plain autoencoder leads that task. The main limitation is interpolation across a topology change: no component of the regularised latent holds both a single-component and a post-break-up shape, whereas an unregularised autoencoder mixes them freely.

1. Introduction

Sparse recordings and costly simulations leave intermediate droplet states unavailable, even when similar shapes occur elsewhere in a campaign. SHROM combines shape-aware geometry, reduced coordinates, and an inverse map so that latent position can correspond to morphology.

1. Introduction

  • A campaign may contain shapes resembling states missed by any one sparse recording, while simulation is too costly to search an operating envelope.A useful representation must be low-dimensional, invertible, and faithful to shape rather than sampling.Source: §1 · p. 2

Approach

  • SHROM combines an elastic shape-space neighbour graph with an autoencoder to make latent position correspond to morphology, a property reconstruction alone does not ensure.The model reconstructs interfaces while penalising latent pairs that are not neighbours when their shapes are neighbours.Source: Approach · p. 2, §2.6 · p. 6

Demonstration

  • The demonstration targets full closed-interface geometry from actuation parameters, extending prior droplet work on descriptors and structure classes.The contribution list also identifies SRVF interfaces as a learning target and tests invertibility and organisation rather than asserting them.Source: Demonstration · p. 3, Contributions · p. 3

2.1. Elastic shape analysis and the SRVF

  • Elastic shape analysis gives meaningful shape distances but no reduced coordinates, so it cannot by itself generate an intermediate interface.The SRVF represents a curve at the same dimension and makes distances meaningful, but supplies no coordinates to move along.Source: §2.1 · p. 4

2.2. Dimensionality reduction and the invertibility gap

  • POD is compact and invertible but measures sampled-coordinate distance, whereas manifold learning can respect a shape metric but provides no map back to a curve.An autoencoder provides a decoder, but reconstruction constrains training points rather than the latent space between them.Source: §2.2 · p. 4, §2.3 · pp. 4–5

2.3. Autoencoders and the geometry gap

  • A plain autoencoder can reconstruct faithfully while assigning arbitrary latent positions, leaving unobserved intermediate regions without guaranteed plausible shapes.That gap matters because a missing frame lies between observations, where reconstruction error is not evaluated.Source: §2.3 · p. 5

2.4. Graph regularisation of a latent space

  • Neighbourhood regularisation is established, but this method builds its graph in elastic shape space so the imposed latent structure reflects shape similarity.The contribution is the composition of that graph with an autoencoder, not the general idea of regularising a latent with neighbourhoods.Source: §2.4 · p. 5, §2.6 · p. 6

2.5. Multi-component shapes

  • For breakup, the method joins components with a degenerate connector to retain a single closed curve and the SRVF apparatus.This is a modelling choice with a measurable cost; alternatives are set aside for implementation, dimensionality, and reparametrisation reasons.Source: §2.5 · pp. 5–6

3. Methodology

SHROM combines a shape metric, a neighbour graph, and an autoencoder to give shape-similar interfaces nearby latent coordinates while decoding every coordinate back to a contour. Its method handles single- and multi-component interfaces, regularises latent organisation, and specifies diagnostics for evaluating the resulting model.

3.1. Overview

  • SHROM combines an elastic shape metric, a graph over that shape space, and an autoencoder whose decoder maps latent coordinates back to contours.The return path allows contours to be generated at coordinates not present in the observations.Source: §3.1 · pp. 6–7, Figure 1 · p. 7

3.2. Shape representation

  • 19.7% average distance reduction from optimising cyclic shift and rotation, and 32.4% when bounded elastic warping is also allowed, quantify the unaligned metric’s overestimate.The distance gap is smaller within a topology class (14.6%) than across classes (22.2%).Source: §3.2 · pp. 7–8

3.3. Assembling multi-component interfaces

  • For multi-component interfaces, components are joined into one closed path with a zero-width connector and resampled equally to keep smaller satellites visible.The connector separates shapes of differing topology, making interpolation across topology change the method’s principal limitation.Source: §3.3 · p. 8

3.4. The shape-space neighbour graph

  • A weighted k-nearest-neighbour graph is built over SRVF coordinates; because the SRVFs are unit-norm, Euclidean and geodesic distances yield the same neighbour ranking.The downstream results are reported as insensitive to the graph metric choice.Source: §3.4 · p. 8

3.5. The regularised objective

  • The training objective combines SRVF reconstruction error with a graph cross-entropy that attracts graph edges and repels sampled non-edges.The affinity kernel’s min_dist parameter controls locality versus global connectivity; the graph penalty is annealed so reconstruction develops first.Source: §3.5 · pp. 8–9

3.6. Decoding and invertibility

  • The decoder normalises its output to the unit sphere, and inversion of the SRVF followed by scale restoration yields a contour from every latent point.Whether those decoded contours are plausible is evaluated by decoding a regular latent grid.Source: §3.6 · p. 9

3.7. Diagnostics

Effective dimensionality..

The effective-dimension diagnostic describes how many latent directions the decoder actually uses.

Effective dimensionality..

  • The participation ratio of squared decoder-Jacobian singular values counts the latent directions the decoder actually uses.It weights directions by the variance they carry rather than by their amplitude.Source: Effective dimensionality · p. 9
  • Squaring the singular values makes the effective-dimension measure reflect variance carried by each direction rather than amplitude alone.Source: Effective dimensionality · p. 9
  • A latent with nominal dimension d can have an effective dimension well below d.Source: Effective dimensionality · p. 9

Isometry defect..

The two diagnostics assess whether latent neighbourhood structure reflects shape-space structure, from preservation of true neighbours to penalties for false ones.

Isometry defect..

  • kNN structure overlap measures the fraction of each shape’s shape-space neighbours retained among its latent neighbours.It is the direct measure of whether the latent preserves shape neighbourhoods.Source: Isometry defect · p. 9
  • The rank-based penalty measures latent-near points that are far apart in shape space.It is reported alongside structure overlap, not as a substitute for it.Source: Trustworthiness · p. 9
  • A good score on either structure overlap or the rank-based penalty alone does not imply a useful embedding.Source: Trustworthiness · p. 9

Trustworthiness..

The paper defines how to read its metrics and test interpolation, then describes an inkjet-droplet corpus and the two-stage screening used to clean it.

Trustworthiness..

  • Table 1 reports metric ranges and reference points, including the value expected from a model carrying no information.Source: Table 1 · p. 10

3.8. Reading the reported quantities

  • Reconstruction error is mean squared error on unit-norm SRVFs, while random corpus shapes are separated by 1.06 ± 0.55 rad on average.The random-pair distance provides a corpus-scale reference for interpreting geodesic errors.Source: §3.8 · p. 10

3.9. Evaluation protocol

  • Interpolation is tested with held-out anchors and leave-one-out paths, compared against a great-circle reference that is not a claim about physical evolution.Adjacent-pair interpolation uses the straight line between consecutive latent codes to isolate representation cost from path-fitting cost.Source: §3.9 · p. 10
  • Latent midpoints decode to valid shapes near both endpoints, but are not generally Fréchet means or geodesic midpoints.Geodesic, Hausdorff, and Chamfer distances provide complementary checks because they respond differently to pointwise defects, outliers, and systematic offsets.Source: §3.9 · pp. 10–11, Shape error · p. 11

4.1. Data

  • The inkjet corpus samples droplet morphologies across actuation settings, with eight waveform parameters recorded for each frame and four separate sequences held out for interpolation.The corpus design makes morphologies along one droplet’s sparsely recorded trajectory likely to be represented by another actuation setting.Source: §4 · p. 11, §4.1 · p. 11

4.2. Two-stage cleaning

Stage 1: per-component screening..

The paper screens the contour corpus at the component and shape levels, then uses a compact autoencoder whose graph regulariser is built over elastic shape space. It also specifies the interpolation protocol and the competing baselines used in evaluation.

Stage 1: per-component screening..

Stage 2: per-cluster outlier removal..

  • Stage 2 removes shapes beyond the 99th percentile of their cluster’s geodesic distance to its mean, removing 3,049 shapes and leaving 301,539.The percentile is applied per cluster because cluster spreads differ; a fixed percentile still removes the same fraction from each cluster, and an adaptive threshold is left for future work.Source: Stage 2: per-cluster outlier removal · p. 12

4.3. Representation and architecture

  • Contours are resampled into 320-dimensional SRVF vectors; multi-component frames retain only the two largest components, joined by a connector.The fully connected encoder maps 320 → 160 → 40 → 10 → d, with a mirrored decoder; d = 2 is the headline setting and d = 5 is a robustness check.Source: §4.3 · p. 13

4.4. Regularisation

  • Training warms up on reconstruction, then introduces the graph term through a six-phase increase in its weight.The warm-up runs up to 4,000 epochs; early graph phases run for 200 epochs each, and the final phase runs up to 3,000 epochs with early stopping.Source: §4.4 · p. 13

4.5. Neighbour-graph metric

  • Euclidean distance on unit-norm SRVFs gives the same neighbour sets as geodesic distance, and cosine distance is likewise monotone in the shape-space angle.Euclidean and geodesic distances have rank correlation 1.000000 on the corpus; the paper reports no measured outcome difference across five graph metrics.Source: §4.5 · p. 13

4.6. Interpolation protocol

  • Interpolation is evaluated by withholding frames from an acquisition, usually at anchor step 10, with anchors placed on both sides of each topology change.Near-change held-out frames measure damage around the discontinuity, not prediction of the discontinuity itself.Source: §4.6 · p. 13

4.7. Baselines

  • The baselines are POD on SRVF vectors and a geometric autoencoder that encourages a locally isometric decoder through a Jacobian penalty.The latter is a pointwise decoder condition, unlike SHROM’s dataset-wide neighbourhood condition.Source: §4.7 · p. 14, POD · p. 14

POD..

The ablation shows that the graph term chiefly changes latent organisation, not reconstruction or fitted readouts. The study also finds that two latent dimensions suffice for downstream tasks, while data filtering changes what the latent decodes.

POD..

  • POD is the linear SRVF reduction baseline; the geometric autoencoder instead uses a decoder-Jacobian isometry penalty, not dataset-wide neighbourhood regularisation.The comparison follows a three-seed protocol, with differences called significant only when they exceed twice the pooled standard deviation.Source: Geometric-AE · p. 14, §4.8 · p. 14

4.8. Statistical protocol

  • The three-seed spread limits claims to the absence of large effects, not the absence of effects; consistent but sub-threshold orderings are reported as trends.The sample standard deviation is used, and a difference is called significant only above twice the pooled standard deviation.Source: §4.8 · p. 14

5.1. Ablation of the manifold-informed loss

  • 0.781 ± 0.049 versus 0.063 ± 0.059 ARI: adding the graph term raises latent clustering agreement with the shape-space partition from chance level.The comparison uses identical architecture, data, schedule and seed, changing only the regularisation term; regression and interpolation are unchanged within noise.Source: Table 3 · p. 15, §5.1 · p. 15

5.2. Latent dimensionality

  • At d = 5, neff = 1.20 ± 0.02 and 99.2% of latent variance lies in two principal components; d = 2 suffices across reported downstream tasks.The corpus is not itself two-dimensional: dimension estimates vary with scale, and PCA needs eight components for 95% of variance.Source: §5.2 · pp. 15–16

5.3. Effect of data cleaning on the latent

  • Removing 1.00% of shapes in stage 2 changes the decoded latent grid qualitatively: all 216 nodes become plausible, while scalar metrics remain unchanged within seed noise.Latent anisotropy rises from 3.2 to 33.8 after filtering.Source: §5.3 · pp. 18–19, Figure 8 · p. 19

5.4. PoC-1: temporal interpolation

  • Interpolation error is 0.325 rad during deformation versus 0.064 rad when relaxed, a 5.1× gap; the droplet spends most of a sequence relaxed and only a small fraction actively deforming.Leave-one-out is about 1.5× easier than the held-out-anchor protocol.Source: Error reporting by regime · p. 19, §5.4 · p. 19

Error reporting by regime..

Interpolation error depends on the deformation regime, anchor density, and sequence difficulty. Topology-aware schemes outperform a naive spline at a change, but their error saturates at the reconstruction limit.

Error reporting by regime..

  • 5.7 to 8.1× worse: the naive cubic spline has higher error than every topology-aware scheme at anchor step 120 and can decode to self-intersecting curves.The four topology-aware schemes are statistically indistinguishable; the piecewise scheme is lowest in all nine comparisons, but no gap clears the significance threshold.Source: Table 6 · p. 20, Error reporting by regime · p. 20
  • The deforming-phase error is under a third of the 1.06-rad separation between random corpus shapes, while the relaxed-phase error is about a sixteenth; pooling understates difficulty because deformation occupies about 10% of frames.Source: Error reporting by regime · p. 20
  • The hardest sequence is more than twice as difficult as the easiest, with the hardest containing seven transition frames in the shortest usable window.The pooled mean is 0.092 at anchor step 10; every tested variant ranks the sequences in the same order.Source: Table 7 · p. 21, Variation across sequences · p. 21
  • Error falls steeply when anchors are scarce, then saturates: using half the trajectory buys little over using one tenth.At the densest setting, SRVF interpolation between decoded anchors matches the best topology-aware scheme, consistent with a reconstruction-limited floor.Source: Table 8 · p. 21

Variation across sequences..

Across four filmed sequences, interpolation is harder when break-up events are concentrated, and its error reaches a reconstruction-limited floor. Within topology classes, latent interpolation gives plausible intermediate contours; across a component-count change, the jump remains.

Variation across sequences..

  • The hardest sequence has more than twice the error of the easiest and contains seven transition frames in the shortest usable window; all trained variants rank sequences alike.The authors report all four sequences rather than excluding the hardest after observing its difficulty.Source: Variation across sequences · p. 21

Composition of the error..

  • At the densest anchor setting, SRVF interpolation between decoded anchors matches the best topology-aware scheme (0.0885 against 0.0891), indicating that error has reached the reconstruction limit.A straight latent line between consecutive frames removes the fitting problem; away from topology changes, midpoint error is 0.91 to 0.98 of endpoint reconstruction error.Source: Composition of the error · p. 21, Isolation by adjacent-frame… · p. 21

Isolation by adjacent-frame interpolation..

  • Within a topology class, ten predicted intermediate states remain credible; the ten-frame held-out sequence includes a change from one contour to two.At anchor step 120, only 12 of 653 frames are observed; topology-aware schemes remain plausible while the naive spline rings across the discontinuity.Source: Figure 11 · p. 22, Figure 10 · p. 22, Failure regime · p. 22

Failure regime..

  • The pinch-off latent jump remains 620×, 225×, or 71× a typical consecutive step across three regularisation strengths; stronger regularisation compresses but does not remove it.The strongest setting gives the smallest jump but the least plausible path, passing through shapes with no tail.Source: Figure 11 · p. 22, Failure regime · p. 23

5.5. PoC-2: waveform to full contour

  • R^2 = 0.878 ± 0.026: waveform parameters predict the latent from which the decoder returns a full closed contour.The result averages three seeds at d = 2; per-dimension R^2 values range from 0.82 to 0.93, and no variant is separable from another.Source: §5.5 · p. 23

5.6. PoC-3: structure recovery

  • Agreement with the shape-space partition rises from chance level to 0.781 ± 0.049 when the graph term is active.The reported seed-to-seed spread means ARI resolves the large ablation effect but cannot rank small differences between strong models.Source: The ablation · p. 23

The ablation..

SHROM’s graph regulariser improves latent shape organisation, while interpolation error favors reconstruction-focused models. The comparisons also show that ARI’s seed spread limits its use for ranking small differences.

The ablation..

  • Even among the six worst test-set predictions, decoded contours retain the correct topology and tail; errors concern body size rather than morphology.The examples are stratified by test-set error rather than sampled randomly.Source: Figure 13 · p. 24

Cluster quality..

  • At K = 4, the latent partition’s silhouette is 0.859 against 0.632 for the geodesic reference, while latent clustering is 40 to 80× faster.The latent partition is better separated at every K; geodesic k-means requires pairwise great-circle distances in 320 dimensions.Source: Cluster quality · p. 24, Cost · p. 24

Cost..

  • ARI’s seed spread of ±0.04 to ±0.14 resolves the twelve-fold ablation effect but cannot resolve few-percent differences between good models.The study therefore reports ARI as mean ± sd and makes no model or configuration ranking from it.Source: Limitations of the adjusted Rand index · p. 24

5.7. Benchmark against linear and geometric baselines

  • SHROM preserves 70 to 104% more neighbourhood structure than the baselines and recovers the shape-space partition far better, at a four-percent reconstruction cost.At d = 5, SHROM’s ARI is 0.781 ± 0.049 against 0.057 ± 0.019 for Geometric-AE.Source: §5.7 · pp. 24–25, Table 9 · p. 25

Transfer of the reconstruction advantage to interpolation..

  • At anchor step 10, Geometric-AE (0.0776±0.0020) and plain autoencoder (0.0784 ± 0.0029) beat SHROM (0.1005 ± 0.0138) on interpolation error.The protocol saturates at the reconstruction limit, ordering models by reconstruction accuracy rather than latent organisation.Source: Transfer of the reconstruction… · p. 25, Table 10 · p. 26
  • At d = 5, the graph term costs approximately 0.022 rad of interpolation error while improving neighbourhood preservation and clustering; its seed spread is 4.8× the plain autoencoder’s.The interpolation cost is about two percent of the 1.06 rad separating two randomly drawn shapes.Source: Transfer of the reconstruction… · p. 26
  • Waveform-parameter regression gives R2 = 0.878 ± 0.026 for SHROM at d = 2, but neither regression gap against the baseline clears twice the pooled standard deviation.At d = 5, the baseline reports R2 = 0.804 ± 0.087 against SHROM’s 0.868 ± 0.010.Source: Transfer of the reconstruction… · p. 26

Similarity graph versus additional capacity..

The graph’s shape-based neighbourhood signal supplies latent organisation that additional capacity or a smoother decoder does not. Metric choice has little measured effect, while regularisation strength trades interpolation against clustering and cannot eliminate the pinch-off discontinuity.

Similarity graph versus additional capacity..

  • Geometric-AE spreads points more evenly, but 77.8% of its grid lies far from training points versus 98.1% for SHROM, whose graph term deliberately clusters shapes.Even spreading and meaningful shape structure are distinct properties.Source: Grid occupancy, where the baseline leads · p. 27

5.8. Variants

  • Rarity-weighted reconstruction yields no measurable task improvement, while a wider decoder’s nominal interpolation gain is not significant and costs four times the decoder parameters.The narrower decoder is retained for parsimony.Source: §5.8 · p. 27

5.9. Sensitivity to the neighbour-graph metric

  • None of 52 comparisons across five graph metrics reaches significance; Minkowski variants change roughly half the graph edges without changing downstream task results.The Minkowski graphs share only 39% and 50% of their edges with Euclidean, yet do not perform worse.Source: §5.9 · p. 27, Table 11 · p. 27
  • Three seeds support ruling out a large graph-metric effect, not ruling out any effect; the seed spread is comparable to every tested gap.Euclidean is retained for parsimony, not superiority.Source: §5.9 · p. 28

5.10. Regularisation strength and the locality–globality trade-off

  • Interpolation error falls from 0.125 to 0.114 as min_dist increases, while clustering favors low min_dist and regression is insensitive to the choice.The interpolation trend reproduces across seeds, but its best-versus-baseline gap is only 1.5 pooled standard deviations.Source: §5.10 · p. 28, Table 12 · p. 28
  • At the most continuous setting, the pinch-off latent jump falls from 513× to 60× a normal step but remains a discontinuity of two orders of magnitude.The result points to separation in the representation that regularisation compresses but cannot remove.Source: Effect of a more continuous latent · p. 28

6. Discussion

The discussion shows that graph regularisation gives latent position morphological meaning, a property reconstruction error cannot measure. It also locates a topology-change boundary that interpolation cannot cross and weighs alternatives, null results, and a competing isometry penalty.

6.1. The role of the manifold-informed loss

  • The graph term leaves reconstruction and regression unchanged but makes latent clustering recover shape-space structure by more than thirteen pooled standard deviations.A plain autoencoder fills four fifths of its decoded grid with the same oval, leaving its latent partition indistinguishable from chance.Source: §6.1 · p. 29

6.2. The central limitation: topology change

  • Topology classes sit 33 times farther apart than typical within-class spacing, and no tested graph component contains both single-component and bridged shapes.The separation persists across four regularisation strengths and four latent dimensions.Source: §6.2 · p. 29
  • The fitted paths are well behaved, but the latent region between topology classes contains no valid intermediate shapes; increasing min_dist compresses, but cannot remove, the discontinuity.The setting that compresses it most produces the least plausible path.Source: §6.2 · p. 29
  • Separate models for each topology class interpolate roughly seven times worse, while a plain autoencoder mixes both families in one component.A trajectory crosses the divide partway through, so no single-topology model sees a whole sequence.Source: §6.2 · p. 29

6.3. Alternatives considered

  • Elastic shape graphs, extended SRVFs, varifolds, and Wasserstein–Fisher–Rao distances each face a stated transfer or metric trade-off for planar interfaces.The paper also quantifies connector distortion inside an elastic metric as a characterisation, not a solution.Source: §6.3 · p. 30

6.4. Negative results

  • Rarity weighting improves nothing, while a wider decoder’s nominal interpolation gain is not significant; the plain autoencoder wins interpolation because error saturates at reconstruction fidelity.Straight-line interpolation between consecutive frames costs no measurable accuracy.Source: §6.4 · p. 30
  • None of 52 comparisons across five graph metrics reached significance against Euclidean distance, including metrics producing graphs with under half their edges in common.The practical condition supported is building the graph over a space where proximity means shape similarity.Source: §6.4 · p. 30

6.5. Suitability of an isometry penalty

  • The isometry baseline reconstructs several times better but recovers shape structure several times worse, consistent with its local Jacobian constraint missing dataset-wide relationships.Its penalty preserves ambient Euclidean geometry although SRVFs lie on a sphere with arc-length distance.Source: §6.5 · pp. 30–31, Isometry with respect to the wrong… · p. 31

Isometry with respect to the wrong geometry..

SHROM combines shape-aware geometry with a decodable, low-dimensional latent, but its purely geometric representation does not bridge topology changes. The discussion reports compact latent use, leaves transfer untested, and distinguishes the interpolation result from what its score measures.

Isometry with respect to the wrong geometry..

  • The isometry penalty targets decoder conditioning, a property close to independent of the property required by a generative shape model.The discussed geometry-preserving alternatives would come at the cost of reparametrisation guarantees relied on here.Source: Isometry with respect to the wrong… · p. 31, §7 · p. 33

6.6. Latent compression relative to the data dimension

  • A wider decoder concentrates latent use further, an observation reported without a proposed mechanism.The concentration is attributed to the trained decoder, not to a two-dimensional corpus.Source: §6.6 · p. 31

6.7. Limits of a purely geometric representation

  • The geometric model imposes no fluid-mechanics constraints, so intermediate states are limited by resemblance to corpus shapes rather than physical admissibility.The paper proposes either physics-informed training penalties or constrained path fitting, but attempts neither.Source: §6.7 · p. 31

6.8. Generality

  • Transfer is a design claim, not a demonstrated result; the recipe is proposed for shape families with a metric representation and a latent intended for sampling.The formulation requires a graph over a shape-similarity space, not a particular distance function.Source: §6.8 · p. 32

7. Conclusions

  • SHROM combines an SRVF shape metric, a neighbour graph, and an autoencoder to decode latent positions into interfaces whose morphology corresponds to position.Removing the graph leaves reconstruction unchanged but makes latent clustering no more informative than chance.Source: §7 · p. 32

6.6. Latent compression relative to the data dimension

  • The learned representation uses two dimensions although the shape corpus has higher intrinsic dimension, allowing the morphology family to be drawn on one plane.A five-dimensional latent uses about one and a fifth of its dimensions and concentrates almost all variance in two.Source: §6.6 · p. 31, §7 · p. 32

7. Conclusions

  • No tested regularised-latent component spans single-component and post-break-up shapes, and separate topology models interpolate substantially worse.The separation is attributed to requiring latent position to track shape similarity, rather than to the connector or embedding dimension.Source: §7 · p. 32
  • A plain autoencoder predicts withheld frames more accurately, but straight-line interpolation costs no measurable accuracy, locating the residual in reconstruction fidelity.The task demonstrates plausible droplets at intermediate times, not the stronger promise implied by its headline number.Source: §7 · p. 33

Appendix A. Supplementary material

The supplementary analyses characterize sampling coverage and latent trajectories, then test connectivity, geometry diagnostics, and regularisation strength. They show that graph-regularised latent components remain topology-pure, while geometry diagnostics do not distinguish graph metrics.

Appendix A.1. Cluster atlas

  • No cluster is predominantly defective, so filtering removes individual shapes by distance from a cluster mean rather than discarding whole clusters.Source: Appendix A.1 · p. 36

Appendix A.2. Coverage of the sampled parameter space

  • Parameter entropy efficiency ranges from 0.79 to 1.00 across eight waveform parameters, and shape-space occupancy is 0.843.Two clusters have only eight members each; these are rare morphologies rather than sampling gaps, so generating there is still extrapolation despite nominal parameter coverage.Source: Appendix A.2 · p. 36

Appendix A.3. Latent trajectories

  • The four filmed sequences form paths through the latent, with topology changes visible as jumps.Source: Appendix A.3 · p. 36

Appendix A.4. Latent connectivity

  • Across four min_dist settings and four latent dimensions, no graph-regularised component contains both single-component and bridged shapes; the plain autoencoder mixes them.At min_dist = 0.3 and 0.8, the largest component is the full single-component class, which is 61.3% of the sample; raising min_dist reconnects fragments within a topology class but not across classes.Source: Appendix A.4 · pp. 36–37, Table A.13 · p. 36

Appendix A.5. Latent geometry diagnostics

  • Across four graph-metric arms, only one of 18 latent-geometry comparisons crossed the two-pooled-standard-deviation threshold, the rate expected by chance.Seed spread exceeded every between-arm difference, so these diagnostics did not rank configurations; connectivity instead separated them by a factor of six.Source: Appendix A.5 · p. 37

Appendix A.6. Regularisation-strength study

  • The continuous min_dist setting fills the latent evenly, while the manuscript setting concentrates it on a strongly anisotropic axis; both decode to plausible shapes throughout.Source: Appendix A.6 · p. 37

Appendix A.7. Reproducibility

  • Dataset construction is deterministic from raw frames and recorded configuration; reported values average seeds 0, 1, and 2 with sample standard deviations.The raw contour dataset is public.Source: Appendix A.7 · p. 37
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