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Walking the Score Manifold: Continuous-time Generative Dynamics on Learned Data Manifolds

Jan Tauberschmidt, Brian B. Moser, Stanislav Frolov, Andreas Dengel, Andrew B. Duncan, Sebastian J. Vollmer

arXiv:2609.17901v1cs.LGcs.AI

TL;DR

Discrete-time generative models tie dynamics to observed timestamps, while continuous-time parameterization alone leaves intermediate evolution underspecified. This paper uses score-based geometry to learn stochastic vector fields along interpolation paths, with simulation-free training and path-relative transverse correction. Experiments span natural video and scientific dynamics, and the authors conclude that score-induced geometry can organize continuous-time generation across these settings.

  • Problem

    Discrete temporal grids tie learned dynamics to observed timestamps, while continuous-time models still lack direct supervision between them and Euclidean interpolants can cross implausible off-manifold regions.

    Source: §1 · p. 1

  • Method

    The framework learns a stochastic vector field along score-induced interpolation paths, using simulation-free local regression and a transverse correction for path-relative robustness.

    Source: §1 · p. 2, §3.2 · p. 5

  • Results

    The framework supports arbitrary-time generation and temporal super-resolution, with demonstrations on natural video, PDE-governed spatiotemporal fields, and molecular dynamics.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Across natural video, PDE-governed dynamics, and molecular trajectories, experiments illustrate that score-induced geometry can organize continuous-time generation beyond task-specific architectures.

    Source: §7 · p. 9

    The method cannot guarantee recovery of true intermediate dynamics; experiments focus on low-resolution settings and moderate training horizons.

    Source: §6 · p. 9

Abstract

from arXiv · show

Generative modeling of time-dependent data is typically formulated on a discrete temporal grid, restricting supervision to the observed timestamps in the training data. We instead frame generation as continuous-time evolution on a learned data manifold. To this end, we leverage pretrained score-based models as geometric priors and learn a vector field that evolves data along score-induced interpolation paths. Because these dynamics follow transitions that respect the geometry learned by the score model, they support generation at arbitrary timestamps and temporal super-resolution beyond the discretization of the training data. Moreover, this geometric formulation allows us to train the vector field simulation-free through a regression objective. To improve long-horizon rollout robustness, we introduce an objective that promotes path-relative transverse exponential stability. While motivated by stability theory, it admits a practical interpretation as denoising score matching transverse to the interpolation path. Further, we extend the framework to a probabilistic setting that models a distribution over plausible future trajectories. We demonstrate the method on natural video and scientific dynamical data, including temporal super-resolution, PDE-based spatiotemporal fields, and molecular dynamics. Our results show that score-based priors provide a strong foundation for learning stochastic continuous-time generative dynamics.

1 INTRODUCTION

Discrete-time training ties learned dynamics to observed timestamps, while continuous-time parameterization alone leaves intermediate evolution underspecified. The paper uses score-informed geometry to learn stochastic continuous-time dynamics that can generate at arbitrary timestamps.

1 INTRODUCTION

  • A fixed temporal grid ties learned dynamics to the resolution of observed data, and continuous-time parameterization alone does not specify how states evolve between timestamps.Simple Euclidean interpolants can also pass through implausible off-manifold regions.Source: §1 · p. 1
  • Score-based priors encode data-manifold geometry that can guide manifold-aware interpolation instead of Euclidean straight lines.The paper frames generation as walking the score manifold in continuous time.Source: §1 · p. 1
  • The proposed stochastic framework learns a history-conditioned vector field whose tangential motion follows score-induced paths and whose transverse correction encourages prior-plausible states.The model supports arbitrary-time generation, simulation-free local regression, and distributions over plausible future trajectories.Source: §1 · p. 2
  • The contributions combine latent-conditioned continuous-time evolution, simulation-free velocity matching with a CVAE likelihood interpretation, and path-relative transverse robustness.The transverse objective corresponds to exponential return toward the interpolation path and has a denoising-score-matching interpretation.Source: §1 · p. 2

2 SCORE-BASED MODELS AS INTERPOLATION PRIORS

The section uses score-based geometry to define paths between data points and an amortized interpolator to approximate them efficiently. The path objective also determines traversal speed.

2 SCORE-BASED MODELS AS INTERPOLATION PRIORS

  • Score-based models encode data-manifold geometry, including local tangent or normal structure, which can be represented with a Riemannian metric.The metric is evaluated at a small positive noise level because score geometry may be poorly behaved at the clean-data limit.Source: §2 · p. 2
  • The metric defines geodesic paths between data points, while an amortized neural correction augments linear interpolation to approximate those paths efficiently.The correction is trained by minimizing metric energy and yields fast, differentiable interpolation that respects score-model geometry.Source: §2 · p. 2
  • The energy objective also determines path traversal, with equality at the minimum exactly when the path has uniform metric speed.The normalized metric arc length covered by time t describes the fraction of the path traversed.Source: §2 · p. 2

3 METHOD: GENERATIVE DYNAMICS ALONG INTERPOLATION PATHS

The method turns score-induced interpolation paths into local training targets for a continuous-time vector field, then extends the model probabilistically with a latent variable. Its local variational objective also supports a sequence-likelihood lower bound under a discretized setup.

3 METHOD: GENERATIVE DYNAMICS ALONG INTERPOLATION PATHS

  • Score-induced interpolation paths provide local evolution targets at multiple time scales for a vector field that can be queried continuously.The velocity model is conditioned on step size and trained by local regression, without backpropagating through an ODE solver.Source: §3 · p. 3

3.1 STATISTICAL FRAMEWORK

  • A latent variable models ambiguity when the same observed history admits multiple plausible future continuations.The model assumes the latent absorbs past dependence once the current state and latent are given.Source: §3.1 · p. 3
  • The probabilistic formulation treats local trajectory evolution as a CVAE, using a full-trajectory posterior and a history-conditioned latent prior.Its reconstruction term becomes squared velocity matching, while the KL term aligns the posterior with the prior.Source: §3.1 · p. 4
  • Under the fixed discretization setup, the local variational objective lower-bounds sequence log-likelihood up to scaling by K and additive constants.The continuous-time latent-driven ODE has zero quadratic variation, so the standard path-space lower-bound machinery does not directly apply.Source: §3.1 · p. 4

Appendix C.1.

The stability construction preserves motion along an interpolation path while contracting off-path deviations, then turns that geometry into a practical regression target. The section also gives a bound for learned-field approximation error and interprets the transverse correction as denoising score matching.

Appendix C.1.

  • The ideal path-relative field is exponentially stable within the tubular neighborhood, returning transverse displacement toward the reference path at rate λ up to longitudinal exit.Its tangential term preserves path motion, and its transverse term contracts off-path displacement.Source: §3.2 · pp. 4–5
  • A learned field with transverse error ε⊥ remains within an exponentially shrinking tube around the path, with residual radius ε⊥/λ.The bound applies when ε⊥ < λρ and until possible longitudinal exit from the tubular neighborhood.Source: §3.2 · p. 5
  • The regression target perturbs a path point normally, then selects an updated target at the required contracted distance that is closest to the uncorrected secant target.The construction is conditioned on z to localize the relevant future path.Source: §3.2 · p. 5
  • At h = 0, the transverse correction coincides with normal denoising score matching, combining tangential velocity matching with transverse denoising.The interpretation treats normal perturbation as Gaussian noise in the normal bundle.Source: §3.2 · p. 5
  • Without latent disambiguation, nearby distinct paths can receive inconsistent contraction targets, pulling states into the wrong basin.The section identifies this as a possible mode-collapse failure and probes it in a 2D example.Source: §3.2 · p. 5
  • The final training stage freezes the score prior and amortized interpolator while training the latent posterior, history-conditioned prior, and step-conditioned velocity model.Time points, step sizes, and noise scales are sampled as design choices.Source: §3.2 · p. 6

4 EXPERIMENTS

Experiments test score-guided continuous-time generation on synthetic paths, PDE fields, molecular trajectories, and natural video. Across these settings, the method shows plausible intermediate states, robust rollouts, and competitive video extrapolation, while the paper notes limits on recovering true intermediate dynamics and scaling.

4.1 DEMONSTRATIVE EXAMPLE

  • In the two-dimensional branching example, the full field transports the domain onto the conditioned path; without correction or noise, off-path states circulate, and without the latent, contraction misses a unique path.The experiments also report plausible stochasticity under partial conditioning and quantify manifold adherence, branching accuracy, and contraction rates.Source: §4.1 · pp. 6–7

4.2 ROLLOUT STABILITY FROM TRANSVERSE CONTRACTION

  • On a single synthetic Navier–Stokes sequence, transverse robustification improves 64-frame rollout stability across direct-step, fixed-step, and adaptive ODE inference.Measurements also verify the desired transverse contracting properties for both pointwise diagnostics and empirical decay rates.Source: Table 1 · p. 6, §4.2 · p. 7

4.3 PLAUSIBILITY OF INTERMEDIATES

  • Score-guided paths provide a physically better Gray–Scott baseline than linear interpolation and keep MD ethanol nearer true geometry, unlike implausible intermediate states from linear-path and Neural ODE fields.On Gray–Scott, the Neural ODE tracks true states most closely but is least plausible spectrally; on MD, competing fields shrink bond lengths to about one-third of their true length.Source: §4.3 · pp. 7–8, Figure 3 · p. 7

4.4 GENERATION OF PDE-GOVERNED SPATIOTEMPORAL DYNAMICS

  • On partially observed Gray–Scott data, model samples remain about as far from the reference set as the reference set is from itself, through four times the training horizon.At T = 64, the normalized sliced Wasserstein distance is close to one; multiple latent samples produce distinct yet plausible futures from the same observation.Source: §4.4 · p. 8

4.5 NATURAL VIDEO GENERATION

  • On KTH extrapolation, our method surpasses Vid-ODE on SSIM, PSNR, and LPIPS, while remaining competitive on interpolation.Vid-ODE drifts off the data manifold over the extrapolation horizon and produces ghosting at unsupervised interpolation times.Source: §4.5 · p. 9

5 RELATED WORK

  • Related work typically supervises continuous-time models only at observed timestamps; this work uses score-induced interpolation targets and path-relative stability rather than generic flow penalties.The extended overview situates the approach within temporal generative modeling, continuous-time dynamics, score-based geometry, and stability.Source: §5 · p. 9, A BACKGROUND: EXTENDED RELATED WORK · p. 12

6 LIMITATIONS

  • The experiments do not guarantee recovery of true intermediate dynamics; they focus on low-resolution settings and moderate training horizons, and scaling may require further engineering.The probabilistic formulation uses a single latent vector, which may not capture semantics at multiple levels.Source: §6 · p. 9

7 CONCLUSION

  • The conclusion presents score-induced geometry as a way to organize continuous-time generation across natural video, PDE-governed dynamics, and molecular trajectories.The framework uses score-based priors for intermediate-time supervision and a simulation-free objective for stochastic vector fields.Source: §7 · p. 9

A.1 DISCRETE-TIME GENERATIVE MODELING OF TEMPORAL DATA

This section contrasts fixed-grid temporal generation with continuous-time, score-informed dynamics learned through simulation-free regression, including stochastic futures and robust rollouts. It also describes finite-step training with a clean-path secant target and velocities rescaled to the physical time interval.

A.1 DISCRETE-TIME GENERATIVE MODELING OF TEMPORAL DATA

  • Fixed-grid temporal generators remain tied to observed timestamps, whereas this work treats evolution between observations as a continuous-time dynamics problem.Prior video methods include autoregressive predictors and joint-frame diffusion or clip generators; intermediate times are generally implicit or interpolated.Source: §A.1 · p. 13

A.2 CONTINUOUS-TIME SEQUENCE AND GENERATIVE MODELING; A.5 FLOW MATCHING AND SIMULATION-FREE GENERATIVE FLOWS

  • The method learns a continuous-time vector field through simulation-free regression on local targets from score-guided interpolation paths, without differentiating through ODE solves.Unlike static-distribution transport, it learns latent-conditioned temporal dynamics from observed states.Source: §A.2 · p. 13, §A.5 · p. 14

A.3 CONTINUOUS-TIME VIDEO GENERATION

  • Vid-ODE (Park et al., 2021) and related latent differential-equation models support arbitrary-time video queries, but lack score-guided, manifold-aware supervision of intermediate observation-space states.The proposed supervision operates in data space or, in KTH experiments, the latent space of a frozen frame autoencoder.Source: §A.3 · p. 13

A.4 SCORE-BASED GEOMETRY AND MANIFOLD-AWARE INTERPOLATION

  • Score-based geometry motivates the interpolation prior: the method combines a score-induced Riemannian metric with an amortized neural interpolator to supervise temporal dynamics.Prior work connects score Jacobians with manifold geometry and uses the induced metric for manifold-aware interpolation (Saito & Matsubara, 2025; Kapu´sniak et al., 2024).Source: §A.4 · pp. 13–14

A.6 STOCHASTIC TEMPORAL PREDICTION AND MULTIMODAL FUTURES

  • The probabilistic formulation couples latent stochasticity to a score-guided continuous-time vector field, modeling multimodal futures with geometry-aware supervision of intermediate states.This addresses a gap in stochastic video and continuous-time predictors, which model distributions over possible futures but do not provide this manifold-aware intermediate-state supervision.Source: §A.6 · p. 14

A.7 STABILITY, CONTRACTION, AND ROBUST ROLLOUTS

  • The robustness objective supervises corrective directions relative to score-guided interpolation paths, yielding path-relative stability tailored to learned manifold geometry rather than generic contraction or smoothness.It is related to contraction analysis but differs from regularizers that indirectly constrain infinitesimal flow properties.Source: §A.7 · p. 14

A.8 SCIENTIFIC DYNAMICAL DATA

  • Scientific experiments motivate the framework because plausible intermediate states can matter alongside endpoint accuracy, while the contribution is a general continuous-time generative framework rather than a domain-specific model.The section names PDE-governed fields and molecular systems as application areas.Source: §A.8 · p. 15

B BACKGROUND: SCORE-INDUCED MANIFOLD; B.3 SCORE-INDUCED RIEMANNIAN METRIC; B.4 GEODESICS ON THE SCORE MANIFOLD; B.5 TRAINING OF THE INTERPOLATOR

  • The score Jacobian defines a metric that makes tangent motion relatively inexpensive and normal motion costly; an amortized neural interpolator approximates low-energy geodesics by minimizing metric energy.Clean endpoints are lifted to a denoising time for geometric computation, and the learned correction augments linear interpolation in lifted space.Source: B BACKGROUND: SCORE-INDUCED MANIFOLD · p. 15, §B.3 · p. 18, §B.4 · p. 19

B.5 TRAINING OF THE INTERPOLATOR

  • Finite-step targets use a secant formed by evaluating the clean path at two nearby interpolation times.The passage specifies a positive time offset h for the nearby evaluation.Source: §B.5 · p. 20
  • Normalized and physical velocities differ by a factor of (∆t_k)^−1, which the main text absorbs into the local target ∆hX_t.Source: §B.5 · p. 20

B.6 UNIFORM METRIC SPEED AND PHASE OF THE INTERPOLATOR

The appendix analyzes the interpolator’s path geometry and traversal phase, establishes transverse contraction along the reference path and in a thin tubular neighborhood, and gives a discretized likelihood lower bound for the discrete model.

B.6 UNIFORM METRIC SPEED AND PHASE OF THE INTERPOLATOR

  • Within reparameterization-closed path families, minimizing metric energy selects a shortest admissible image traversed at uniform metric speed; the objective recovers linear interpolation for the flat metric.The time profile can also be changed by monotone warping without changing the path image or length.Source: §B.6 · pp. 21–22

C.1 ELBO AND DISCRETIZED LIKELIHOOD LOWER BOUND

  • For a uniform discrete grid, Proposition 1 establishes a discretized likelihood lower bound, and the proposed objective adds K −1 KL terms while allowing arbitrary conditioning.The passage relates this objective to Posterior Matching (Strauss & Oliva, 2022).Source: §C.1 · p. 23

C.2 EXPONENTIAL DECAY OF TRANSVERSE DISPLACEMENT

  • Under the tubular-neighborhood assumption, the target field is path-relative transversely exponentially stable at rate λ until possible longitudinal exit from the tube.The transverse displacement is non-increasing, so exit through the tube’s lateral boundary is excluded during the bound’s interval of validity.Source: §C.2 · pp. 25–26

C.3 TRANSVERSE CONTRACTION

  • Under Proposition 2’s assumptions, the path-relative target field is transversely contracting after the tubular neighborhood is possibly shrunk.Source: §C.3 · p. 28
  • The vector field v⋆ is transversely λ-contracting on reference path Γ.Source: §C.3 · p. 28
  • Because v⋆ is C1 near compact Γ and nonzero on Γ, a sufficiently thin tubular neighborhood satisfies the transverse λ/2-contraction inequality.Source: §C.3 · p. 28

C.4 APPROXIMATION ERROR AND TUBE STABILITY

This appendix bounds how transverse approximation error affects trajectories near the reference path and derives the finite-step correction used to preserve transverse contraction. It also records the scope and compute budget of the experiments.

C.4 APPROXIMATION ERROR AND TUBE STABILITY

  • With transverse error ε⊥<λρ, approximate trajectories cannot exit the tube laterally; if they remain there indefinitely, their transverse displacement is bounded by ε⊥/λ.The first possible exit is longitudinal; the resulting tube radius is time-uniform.Source: §C.4 · pp. 29, 31
  • Contraction balances linear transverse approximation error against quadratic displacement, and the bound depends only on the field error's normal component.The correction upgrades unbounded linear-in-time error growth to a time-uniform tube, with larger λ tightening the tube but stiffening the target field.Source: §C.4 · pp. 30–31

C.5 SECANT CORRECTION FOR FINITE-STEP TARGETS

  • The unique minimal finite-step correction makes transverse displacement decay by exp(−λh) relative to the next secant endpoint.Choosing perturbations orthogonal to the secant preserves the longitudinal step, unlike tangent-normal perturbations on curved paths.Source: §C.5 · pp. 31–32

D DATASETS AND TRAINING DETAILS

  • The reported full generative-model runs used at most three days per run; auxiliary flow-matching and interpolator training for 2D, PDE, and molecular experiments each used at most one day.KTH latent-pipeline stages each trained for roughly one to two days on a single GPU.Source: D DATASETS AND TRAINING DETAILS · p. 33

D.1 IMPLEMENTATION DETAILS: FULL TRAINING ALGORITHM

The implementation samples local interpolation targets and trains the vector field by regression, optionally adding transverse robustness and latent conditioning. Backbone architectures are selected to suit each data representation.

D.1 IMPLEMENTATION DETAILS: FULL TRAINING ALGORITHM

  • Training samples adjacent endpoints, interpolation times, and step sizes from a frozen prior, then regresses the vector field to local targets with an optional KL term.The KL gradients are usually stopped from flowing into posterior parameters, so KL acts as prior matching rather than directly contracting the posterior.Source: §D.1 · pp. 33–34
  • The optional robustness objective perturbs inputs orthogonally to the tangent or secant and regresses to corrected targets; setting noise scale σk=0 disables it.For h=0 the perturbation probes tangent-transverse directions; for h>0 it is orthogonal to the secant.Source: §D.1 · p. 33

D.2 IMPLEMENTATION DETAILS: NEURAL ARCHITECTURES

  • Backbones vary by representation: residual MLPs model 2D trajectories, U-Nets model PDE fields, and an equivariant graph network models molecular coordinates.The KTH latent-video model uses a related U-Net with spatial-grid latent conditioning.Source: §D.2 · pp. 34–35
  • For molecular trajectories, the EGNN updates coordinates using relative displacement vectors, preserving translation and rotation equivariance.Its messages use atom features and squared interatomic distances.Source: §D.2 · p. 35

D.3 DEMONSTRATIVE EXAMPLE

The demonstrative example tests whether the model follows a branched square manifold and captures future branch choices from partial histories. Ablations compare score-induced paths, target correction, and latent conditioning.

D.3 DEMONSTRATIVE EXAMPLE

  • The square manifold uses inner and outer cubic Bézier arcs, while training sequences expose sparse points on closed loops with randomized phase and branch choices.The score prior sees noisy samples from all eight arcs, whereas the sequence model sees the sparse loop observations.Source: §D.3 · p. 35
  • The ablation evaluates off-manifold distance, branch accuracy, and transverse contraction rate; the latent-free field reaches approximately 1/16 branch accuracy by chance.Each of the twelve configurations changes only listed components and uses one training run.Source: §D.3 · pp. 35–36

D.4 KTH ACTION

  • The KTH action subsection documents the configuration behind Figure 4, Figure 11, and Tables 5 and 6.Source: §D.4 · p. 36

D.4.1 DATA AND EVALUATION PROTOCOL

The KTH appendix specifies the data split and evaluation protocols, then details a three-stage latent-video pipeline and its inference procedure. A retrained Vid-ODE baseline closely reproduces published interpolation metrics.

D.4.1 DATA AND EVALUATION PROTOCOL

  • The 599-video KTH dataset uses a 16/9 subject split, with 383 training and 216 test videos; interpolation and extrapolation are scored on distinct frames from the first 10.Predictions are quantized to 8 bits before scoring, and the metrics follow Vid-ODE’s protocol and implementation.Source: §D.4.1 · p. 36

D.4.2 STAGE 1: FRAME AUTOENCODER

  • The frame autoencoder is trained on individual KTH training frames, frozen for later stages, and reaches approximately 42 dB PSNR on test-set reconstruction.Its latent representation has shape 32×32×4.Source: §D.4.2 · pp. 36–37

D.4.3 STAGE 2: SCORE MODEL AND INTERPOLATION PATH

  • The score model learns flow matching from Gaussian noise to latent frames, while a learned interpolator constructs paths between consecutive latents in score-lifted space.The endpoints are lifted and denoised using the score model’s probability-flow ODE; the linear ablation substitutes plain latent interpolation.Source: §D.4.3 · p. 37

D.4.4 STAGE 3: LATENT VIDEO-GENERATION MODEL

  • A conditional latent velocity field is trained over length-10 windows, with a latent variable encoding sequence-level factors and the prior conditioned on protocol-observed frames.The prior observes odd-indexed frames for interpolation or the first five frames for extrapolation; the posterior sees the full window.Source: §D.4.4 · pp. 37–38, Table 9 · p. 38

D.4.5 INFERENCE

  • Inference samples one latent from the prior and uses an explicit Euler rollout at four times the native frame rate before decoding scored frames.The rollout starts at frame 1 for interpolation and frame 5 for extrapolation.Source: §D.4.5 · p. 38

D.4.6 VID-ODE BASELINE

  • The retrained Vid-ODE interpolation baseline closely matches published SSIM and LPIPS and exceeds published PSNR by 1.3 dB.SSIM is 0.912 versus 0.911 published, LPIPS 0.049 versus 0.048, and PSNR 33.08 versus 31.77.Source: §D.4.6 · p. 38

D.5 PDE DATA

The appendix describes synthetic PDE data, a fine-time reference, a wider-spacing supervision comparison, and a fully observed Gray–Scott physics-supervision experiment. These protocols specify data scope and evaluation conditions, including a held-out fine-trajectory test and a physical-residual objective.

D.5 PDE DATA

  • The PDE experiments use 64×64 scalar fields with 64 stored frames per trajectory, 128 trajectories, and training subsequences of length 16.The datasets are Navier–Stokes and Gray–Scott; the observed Navier–Stokes crop is an open subsystem influenced by surrounding unobserved flow.Source: §D.5 · pp. 39–40

D.6 FINE-TIME GROUND-TRUTH REFERENCE

  • Fine-time plausibility is evaluated on eight fresh Gray–Scott trajectories with unseen intermediate states, comparing interpolation paths against true states at nine interior offsets per segment.Results are reported as mean ± standard deviation over trajectories using relative L2 error, velocity cosine, and a spectral diagnostic.Source: §D.6 · p. 41

D.7 WIDER FRAME SPACING: SUPERVISION COMPARISON

  • The wider-spacing comparison trains score-target and linear-target fields plus a Neural ODE on shared backbones, and verifies the Neural ODE fits training endpoints 13 to 21 times more accurately than the best simulation-free model fits evaluation endpoints.Intermediate states are evaluation-only and segments are teacher-forced from their ground-truth left frames.Source: §D.7 · p. 41

D.8 FULLY OBSERVED GRAY–SCOTT AND PHYSICS SUPERVISION

  • In the two-channel Gray–Scott experiment, V-score trains on metric energy, V-phys minimizes PDE residual to seek dynamically consistent rather than geometrically short paths, and V-lin is an untrained linear path.Evaluation covers all 8 trajectories; the true fine trajectory’s central-difference residual RMS is 1.79 · 10−5 in physical units, quoted as the attainable floor.Source: §D.8 · pp. 41–42

D.9 MOLECULAR DYNAMICS

The molecular-dynamics appendix adds junction continuity and a speed floor to interpolation, then evaluates the augmented method and generative fields on ethanol data. It also states a limitation for unconditional paths: reproducing thermal jitter likely requires conditioning on external forcing.

D.9 MOLECULAR DYNAMICS

  • On ethanol trajectories sampled every 100 native MD steps, observed segments leave 99 unseen raw frames per interval as intermediate ground truth.The prior and interpolator use the first 90% of the trajectory, while the generative fields are fit to one four-frame window.Source: §D.9 · p. 42
  • The augmented interpolator penalizes velocity mismatches at shared frames and adds a speed floor to avoid the trivial stalled-path solution.Its Euclidean speed reference is fixed by observed endpoints, and lifted-space norms are used to constrain the path without inflating the correction beyond the score model’s valid region.Source: §D.9 · p. 43
  • The endpoint-vanishing path parameterization preserves observed frames exactly while allowing velocity continuity to carry from lifted space into clean coordinates.The continuity follows because adjacent segments denoise the same lifted shared point.Source: §D.9 · p. 43
  • The augmented interpolator reduces the shared-frame kink angle from 92.5° to 28.0° and metric energy from 2247 to 930, compared with 8.9° for the true trajectory.The unaugmented kink is close to the 107.2° angle of linear segments.Source: §D.9 · pp. 43–44
  • The score-target and linear-target molecular fields share an autonomous EGNN backbone and differ in whether training targets come from the augmented score interpolator or linear paths.The Neural ODE regresses segment endpoints and was unstable during training, so only its best-loss results are reported.Source: §D.9 · p. 44
  • Unconditional paths likely cannot reproduce small-scale thermal jitter without the external forcing that produced the trajectory.The authors leave force-conditioning for future work.Source: §D.9 · p. 44

D.10 INFERENCE AND FIGURE DETAILS

  • The molecular figure rolls out all three fields freely from the first observed frame and plots O–H bond length and C–O–H bond angle against rollout time alongside the native-resolution trajectory.Each rollout step lands on a raw MD frame, with observed-frame times marked.Source: §D.10 · p. 46

E.1 DEMONSTRATIVE EXAMPLE

The 2D example tests whether a learned field follows score-informed paths through a branching manifold and supports conditional trajectory sampling. Its samples preserve observed branch choices while retaining diversity in unresolved continuations.

E.1 DEMONSTRATIVE EXAMPLE

  • As observed context grows, conditional samples preserve observed arch choices while remaining diverse over unobserved loop sections.The latent distributions are also visualized after projection onto their two leading principal components.Source: §E.1 · p. 46, Figure 5 · p. 46, Figure 6 · p. 47
  • Changing the latent switches the conditioned field to another branch configuration, with each latent corresponding to a single trajectory.The reported ablation table covers removal of individual components, including a correction-removal variant that retains input noise.Source: §E.1 · p. 47, Figure 7 · p. 47, Table 13 · p. 47

E.2 ROBUSTNESS

The corrected objective improves reconstruction rollouts across solver settings and remains stable at coarse steps. It trains the vector field to return perturbed states toward the interpolation path.

E.2 ROBUSTNESS

  • The corrected objective attains consistently lower rollout error across solver settings and remains stable at coarse step sizes.Figure 8 reports mean LPIPS over rollout under three inference regimes.Source: §E.2 · p. 47, Figure 8 · p. 48
  • Without transverse correction, deviations can compound; the robustified objective returns perturbed states toward the interpolation path for more stable long-horizon reconstructions.The authors report that path-relative transverse contraction is necessary to prevent catastrophic error accumulation.Source: §E.2 · p. 48, Figure 9 · p. 48

E.3 PLAUSIBILITY OF INTERMEDIATES

Across scientific interpolation experiments, score-induced paths better match plausible intermediate dynamics than linear paths, while physics-based supervision can further align paths with known equations. Natural-video interpolation and long-horizon Gray–Scott rollouts provide additional qualitative tests of the learned dynamics.

E.3 PLAUSIBILITY OF INTERMEDIATES

  • The score path follows true intermediate motion more faithfully than the linear path, with velocity cosine 0.996 versus 0.965.The paths separate at every interior offset; Table 14 compares frozen interpolation paths with true fine-time intermediates.Source: §E.3 · p. 48, Table 14 · p. 49
  • At w=20, the score-supervised field reaches rel-L2 0.115 versus its frozen teacher’s 0.116 and spectral value 0.335 versus 0.331.Across wider spacings w=15 and w=20, the Neural ODE has the best tracking error and velocity cosine, but its spectral value degrades from 0.303 to 0.482 and is worst at w=20; the score-supervised field stays with its teacher across metrics.Source: §E.3 · p. 49, Table 15 · p. 49
  • On fully observed two-species Gray–Scott data, score geometry improves 2.7× over linear interpolation, while PDE-residual training brings the path within 3× of the ground-truth residual floor.The comparison uses the known governing equations when available and score geometry when they are not.Source: §E.3 · p. 49, Table 16 · p. 49
  • At K=2000, the score-path residual reaches 5.6 · 10−5 (3.1× the true-trajectory residual), versus 1.6 · 10−4 (8.9×) for the linear path.The linear path remains 2.6–2.8 times farther from true states in relative L2 across refinement budgets; refinement does not recover the geometry it missed.Source: §E.3 · pp. 49–50, Table 17 · p. 50
  • In Navier–Stokes sub-frame comparisons, the score-supervised model varies smoothly in enstrophy, while the linear-target model shows sharp changes.Both models capture the coarse transition, but the linear model’s intermediate evolution is visibly less regular.Source: §E.3 · p. 50, Figure 10 · p. 50

E.4 NATURAL VIDEOS: INTERPOLATION

  • At unsupervised half-times on KTH, Vid-ODE shows ghosting on moving limbs, while our rollout remains closer to the true trajectory.The comparison observes every second frame and displays all model outputs on a shared half-frame grid; observed-frame metrics do not expose this failure mode.Source: §E.4 · p. 50, Figure 11 · p. 51

E.5 PDE-GOVERNED SPATIOTEMPORAL FIELDS

  • Conditioned only on the first Gray–Scott concentration field, the latent-conditioned model generates distinct, visually plausible futures up to T=256.Training uses subsequences of length 16, and available training trajectories are at most 64 frames long; the distinct futures reflect ambiguity from the unobserved chemical species.Source: §E.5 · p. 51, Figure 12 · p. 51
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