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Stochastic Liquid Deformation Fields: An SDE Generalisation of Closed-Form Continuous-Time Cells for Dynamic 3D Gaussian Splatting

Mingzhao Li, Arghya Pal

arXiv:2608.28702v1cs.CV

TL;DR

D-3DGS’s closed-form CfC field provides continuous-time deformation but omits the diffusion associated with its stochastic lineage. This paper restores diffusion with a training-only Gaussian perturbation of the time gate, finding parity with deterministic CfC on D-NeRF and no benefit from noise on NeRF-DS.

  • Problem

    The deterministic CfC retains the drift of a noise-driven liquid system but drops diffusion, motivating a solver-free way to restore the stochastic term.

    Source: §I · p. 1

  • Method

    The method adds a small Gaussian increment to each CfC cell’s time-gate pre-activation as a one-step SDE realization, with no solver or added parameters and exact CfC recovery at λ=0.

    Source: §I · p. 1, B. Stochastic (SDE) Time Gate · p. 2

  • Results

    On D-NeRF the stochastic field matches deterministic CfC and is ahead of the MLP, while on NeRF-DS deterministic CfC remains best and constant noise does not help.

    Source: §VI · p. 5

  • Takeaways & Limitations

    The CfC field has a clean SDE interpretation and a free noise knob, but standard interpolation benchmarks favor keeping that knob near zero.

    Source: §VI · p. 5

    The clean, dense D-NeRF and NeRF-DS protocol under-tests corrupted or sparse supervision, where the proposed noise is intended to help.

    Source: §V · p. 5

Abstract

from arXiv · show

Deformable 3D Gaussian Splatting (D-3DGS) reconstructs dynamic scenes by deforming a canonical set of 3D Gaussians through a deformation field of frame time. Replacing its MLP with a stack of Closed-form Continuous-time (CfC) cells-a Liquid Neural Network that solves the Liquid Timeconstant ODE in closed form-gives the field continuous-time behaviour at feed-forward cost. That closed form, however, is only the deterministic limit of a noise-driven system, and drops the stochastic term usually credited for the robustness of liquid networks. We put it back: a small Gaussian perturbation is added to the time gate of every CfC cell, turning the deterministic field into a simple stochastic (SDE) one. The noise is used only during training, needs no solver, and reduces exactly to the CfC when switched off. On the synthetic D-NeRF scenes the stochastic field is on par with the deterministic CfC and beats the MLP baseline on most scenes; on the real-world NeRF-DS scenes the deterministic limit is already best and adding noise does not help. The study thus gives both a clean way to read the CfC field as an SDE and an honest account of when a plain noise term helps and when it does not.

I. INTRODUCTION

The paper replaces the temporally independent D-3DGS MLP with a solver-free CfC field, then restores a dropped stochastic term by perturbing each cell’s time gate. The resulting field is competitive on synthetic scenes, while deterministic CfC remains preferable on real-world scenes.

  • D-3DGS uses a positional-encoded MLP to deform canonical Gaussians into dynamic scenes, but its sampled timesteps are architecturally uncoupled.Source: §I · p. 1
  • CfC cells provide explicit continuous-time semantics through a learned sigmoidal time gate while retaining feed-forward inference without a numerical solver.Source: §I · p. 1
  • The method adds a small Gaussian increment to each CfC cell’s time-gate pre-activation, restoring diffusion without changing the inference path.Source: §I · p. 1
  • At λ=0, the stochastic field exactly reduces to the deterministic CfC, and training-only noise leaves inference unchanged.Source: §I · p. 1
  • On D-NeRF, the liquid fields are competitive with or ahead of the D-3DGS MLP, while on NeRF-DS the deterministic limit remains best.Source: §I · pp. 1–2
  • The noise is harmless on synthetic data but does not help and slightly hurts on real-world data, where deterministic CfC is already difficult to improve.Source: §I · pp. 1–2

II. RELATED WORK

Prior work spans dynamic Gaussian deformation, solver-based continuous-time models, stochastic neural dynamics, and liquid networks. This paper positions its approach as a solver-free stochastic extension of the CfC deformation field.

  • Dynamic Gaussian methods include D-3DGS’s MLP deformation field and explicit-flow or dual-deformation alternatives such as Gaussian-Flow, Shape of Motion, and FLAG-4D.Source: §II · p. 2
  • The paper identifies a gap: existing deformation methods do not treat Gaussian deformation as an explicit stochastic continuous-time function of t.Source: §II · p. 2
  • Neural ODE and Neural SDE approaches model continuous-time dynamics with numerical integration, incurring solver cost during forward passes.Source: §II · p. 2
  • Liquid Time-constant networks use explicit time constants and input-dependent forcing, while CfC analytically solves the associated ODE with a sigmoidal time gate.Source: §II · p. 2

A. Background: Liquid Deformation Field

The liquid deformation field represents canonical Gaussians and predicts time-dependent offsets using a stack of CfC cells. Its cell updates use a shared backbone, linear heads, and a sigmoidal gate, while the surrounding D-3DGS pipeline remains unchanged.

B. Stochastic (SDE) Time Gate

The method restores stochasticity to the CfC by perturbing its time-gate pre-activation with one Gaussian Euler–Maruyama increment. This first-order gate-only approximation preserves solver-free inference and exactly recovers the deterministic CfC when λ=0.

Stochastic gate

Properties

Approximation scope

  • The formulation is a first-order, single-step approximation that perturbs the time response without modifying the candidate states.The authors do not claim a higher-order or hidden-state SDE.Source: B. Stochastic (SDE) Time Gate · p. 2

C. Implementation

The implementation adds one training-time Gaussian perturbation to each cell’s time gate while keeping the deterministic liquid-field architecture otherwise unchanged. Noise is independently sampled per cell and forward pass, with no added parameters or test-time cost.

Architecture

  • The model uses six CfC cells, hidden width 128, backbone depth 2, GELU activation, and time-readout widths of 32 or 64.The time-readout width is 32 for D-NeRF and 64 for NeRF-DS.Source: C. Implementation · p. 2
  • Only λ is newly introduced, while the backbone width matches the deterministic liquid field.Source: C. Implementation · p. 2

Noise implementation

  • Noise is sampled independently per cell and forward pass and disabled at inference.Its training overhead is one Gaussian sample and one addition per cell.Source: C. Implementation · p. 2, Figure 1 · p. 3

IV. EXPERIMENTS

Experiments compare the stochastic field with retrained MLP and deterministic CfC baselines on synthetic D-NeRF and real-world NeRF-DS scenes using PSNR, SSIM, and LPIPS. Results are interpreted with a measured-run variability floor for PSNR differences.

Datasets

  • The evaluation covers eight synthetic D-NeRF scenes and seven real-world NeRF-DS scenes with specular, dynamic objects.Source: A. Setup · p. 3

Protocol

  • The study reports PSNR, SSIM, and LPIPS against the original D-3DGS MLP and deterministic CfC baselines.Baselines use identical training protocols and seeds.Source: A. Setup · p. 3
  • The proposed SDE-LNN uses calibrated noise level λ=0.05 unless otherwise stated.Source: A. Setup · p. 3

Statistical calibration

  • PSNR differences below approximately 0.1 dB are treated as ties because repeated Hell Warrior runs span 41.48–41.57 dB.The reported calibration standard deviation is approximately 0.05 dB.Source: A. Setup · p. 3

B. D-NeRF: Eight Synthetic Scenes

On the eight synthetic D-NeRF scenes, the stochastic liquid field generally outperforms the matched MLP, with gains concentrated on high-frequency articulated motion, while retaining solver-free, size-matched inference.

B. D-NeRF: Eight Synthetic Scenes

  • 38.28 vs. 38.19 dB mean PSNR: SDE-LNN exceeds the matched D-3DGS MLP baseline across the eight D-NeRF scenes.SDE-LNN attains the best PSNR on six of eight scenes, although the aggregate gap is within measured variance.Source: B. D-NeRF: Eight Synthetic Scenes · p. 3
  • Hook gains +1.24 dB over the MLP, with additional gains on Hell Warrior, Stand Up, and Jumping Jacks.The clearest improvements occur on scenes with high-frequency articulated motion.Source: B. D-NeRF: Eight Synthetic Scenes · p. 3
  • SDE-LNN and deterministic CfC are exactly size-matched for isolating noise effects, while the liquid field can be 39% smaller than the MLP on NeRF-DS.All compared fields use feed-forward inference without a numerical solver.Source: D. Compute and Overhead · pp. 3–4
  • The D-NeRF comparison uses PSNR, SSIM, and LPIPS at 800×800, with retrained MLP and SDE-LNN models under identical protocol and seeds.The table compares eight synthetic scenes and distinguishes best, second-best, and third-best cells.Source: Table I · p. 4

E. Noise-level Study

The noise sweep finds no aggregate benefit from constant Gaussian noise: small noise is harmless on D-NeRF, whereas the deterministic CfC is best on NeRF-DS and positive noise slightly degrades it.

E. Noise-level Study

  • No noise level produces an aggregate gain in the dataset-mean sweep.The study compares metrics as a function of λ.Source: E. Noise-level Study · p. 4
  • On D-NeRF, the mean remains unchanged up to λ=0.05, while larger λ gradually washes out real motion and lowers the mean.This identifies λ=0.05 as the largest tested synthetic-data noise level before quality begins to fall.Source: E. Noise-level Study · p. 4
  • On NeRF-DS, λ=0 gives the best PSNR and every positive λ is slightly worse.Here λ=0 is the deterministic CfC limit.Source: E. Noise-level Study · p. 4

F. Qualitative Results

Qualitatively, the largest D-NeRF gains appear around fast-moving articulated limbs, where the stochastic liquid field preserves sharper silhouettes than the MLP.

F. Qualitative Results

  • The stochastic liquid field keeps fast-moving articulated limb edges sharper and closer to the ground-truth silhouette than the MLP.The MLP shows smearing or faint trailing ghosts around the limb.Source: F. Qualitative Results · p. 4

V. DISCUSSION

On dense interpolation benchmarks, deterministic CfC already performs well, so constant gate noise is neutral at best and can hurt real-world results. The SDE formulation remains useful as a solver-free, tunable framework, with adaptive stochasticity left for noisier supervision.

Noise efficacy

  • Dense temporal supervision and interpolation let deterministic CfC fit well, leaving little error for constant noise to correct.On noisier real-world scenes, the noise can do more harm than good.Source: §V · p. 5

Future direction

  • Adaptive stochasticity is proposed for jittery supervision, including fast D-NeRF motion and difficult NeRF-DS scenes.Suggested variants include annealing λ, input-dependent λ, or diffusion in the hidden state.Source: §V · p. 5

Limitations

  • The clean, dense evaluation protocol under-tests corrupted or sparse supervision, such as jittered timestamps, pose noise, and few-view training.The study uses one constant, gate-level λ, and D-NeRF differences fall within reported run-to-run variance.Source: §V · p. 5

Takeaway

  • The SDE view restores diffusion with deterministic inference, but on standard interpolation benchmarks its noise knob is best left near zero.Its supported value is a clean formulation and free tunability rather than an automatic quality gain.Source: §VI · p. 5

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