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

Mingzhao Li; Arghya Pal

cs.CVAug 27, 2026

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

Authors: Mingzhao Li, Arghya Pal

Organizations: School of Information Technology, Monash University, Selangor, Malaysia

Abstract

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.

Figures & tables

Figure 1: Stochastic liquid cell. A shared backbone $\phi([u;h])\!\to\!z$ feeds four heads: $g,h_{\mathrm{cand}}$ form two candidate states, while $W_{a}z,W_{b}z$ form the affine elapsed-time term. The time gate is the only path through which $\tau$ enters the cell; we add a Gaussian diffusion increment $\lambda\varepsilon$ to its pre-activation ( $\sigma_{\tau}=\sigma(W_{a}z\,\tau+W_{b}z+\lambda\varepsilon)$ ), an Euler–Maruyama step of the SDE whose drift is the deterministic gate. Noise is active in training only.

	Hell Warrior			Mutant			Hook			Bouncing Balls
Method	PSNR $\,\uparrow$	SSIM $\,\uparrow$	LPIPS $\,\downarrow$	PSNR $\,\uparrow$	SSIM $\,\uparrow$	LPIPS $\,\downarrow$	PSNR $\,\uparrow$	SSIM $\,\uparrow$	LPIPS $\,\downarrow$	PSNR $\,\uparrow$	SSIM $\,\uparrow$	LPIPS $\,\downarrow$
D-NeRF [ 11 ]	24.06	0.9440	0.0707	30.31	0.9672	0.0392	29.02	0.9595	0.0546	38.17	0.9891	0.0323
TiNeuVox [ 2 ]	27.10	0.9638	0.0768	31.87	0.9607	0.0474	30.61	0.9599	0.0592	40.23	0.9926	0.0416
D-3DGS (MLP) [ 17 ]	41.23	0.9867	0.0255	42.05	0.9944	0.0067	36.94	0.9855	0.0167	41.37	0.9957	0.0086
Ours (SDE-LNN)	41.61	0.9873	0.0259	42.08	0.9941	0.0076	38.18	0.9884	0.0144	40.22	0.9950	0.0100
	Lego			T-Rex			Stand Up			Jumping Jacks

Table 1: Per-scene comparison on the eight D-NeRF synthetic scenes. PSNR

\,(\uparrow)

, SSIM

\,(\uparrow)

and LPIPS

\,(\downarrow)

800\!\times\!800

test images. D-NeRF and TiNeuVox numbers are quoted from [ 17 ] ; D-3DGS (MLP) and Ours (SDE-LNN,

\lambda{=}0.05

) are retrained under an identical protocol with the same canonical-Gaussian set and seeds. Cells are coloured best , second-best , third-best .

	Sieve			Plate			Bell			Press
Method	PSNR $\,\uparrow$	SSIM $\,\uparrow$	LPIPS $\,\downarrow$	PSNR $\,\uparrow$	SSIM $\,\uparrow$	LPIPS $\,\downarrow$	PSNR $\,\uparrow$	SSIM $\,\uparrow$	LPIPS $\,\downarrow$	PSNR $\,\uparrow$	SSIM $\,\uparrow$	LPIPS $\,\downarrow$
NeRF-DS [ 16 ]	25.78	0.8900	0.1472	20.54	0.8042	0.1996	23.19	0.8212	0.1867	25.72	0.8618	0.2047
TiNeuVox [ 2 ]	21.49	0.8265	0.3176	20.58	0.8027	0.3317	23.08	0.8242	0.2568	24.47	0.8613	0.3001
D-3DGS (MLP) [ 17 ]	25.28	0.8721	0.1514	20.50	0.8140	0.2265	25.07	0.8389	0.1637	25.35	0.8618	0.1939
Ours (SDE-LNN)	25.43	0.8690	0.1627	20.43	0.8142	0.2314	25.14	0.8399	0.1702	25.49	0.8643	0.2015
	Cup			As			Basin			Mean

Table 2: Per-scene comparison on the seven NeRF-DS real-world scenes. NeRF-DS and TiNeuVox numbers are quoted from [ 17 ] ; D-3DGS (MLP) and Ours (SDE-LNN,

\lambda{=}0.05

) are retrained under an identical protocol. Colouring as in Table 1 .

Deformation field	Params (M)	Inference
D-3DGS MLP ( $D{=}8,W{=}256$ ) [ 17 ]	0.52	feed-forward
Liquid CfC / SDE-LNN — NeRF-DS	0.32	feed-forward
Liquid CfC / SDE-LNN — D-NeRF	1.55	feed-forward

Table 3: Deformation-field parameter counts, measured with ptflops [ 12 ] on the exact configurations used. The deterministic CfC and SDE-LNN are architecturally identical—the training-time noise adds no parameters and is disabled at inference—so they share a row. The MLP count is the D-3DGS deformation field [ 17 ] .

	D-NeRF (8 scenes)			NeRF-DS (7 scenes)
$\lambda$	PSNR	SSIM	LPIPS	PSNR	SSIM	LPIPS
0.00	38.28	0.9856	0.0168	23.83	0.8482	0.1908
0.05	38.28	0.9856	0.0170	23.73	0.8483	0.1921
0.10	38.06	0.9853	0.0173	23.73	0.8480	0.1912
0.20	37.91	0.9854	0.0175	23.80	0.8489	0.1903

Table 4: Noise-level study: dataset-mean metrics as a function of

\lambda

\lambda{=}0

is the deterministic CfC. best per column.

Panel of Figure 2 — Figure 2: Qualitative comparison on D-NeRF. The three scenes with the largest PSNR gain over the MLP (Table 1 ). Each row is one scene; within a row, left to right: ground truth , D-3DGS (MLP) , Ours (SDE-LNN) . Look at the fast-moving limb in each scene, where the MLP smears or ghosts the motion.

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Liquid Neural Networks as a Drop-in Continuous-Time Deformation Field for Dynamic 3D Gaussian Splatting

Mingzhao Li, Arghya Pal, Guan Yuan Tan

School of Information Technology, Monash University, Selangor, Malaysia

3D Gaussian Splatting Liquid Neural Networks Dynamic Scene Reconstruction Neural Radiance Fields

Open paper AskPDF arXiv

Deformable 3D Gaussian Splatting (D-3DGS) re-constructs dynamic scenes from monocular video by deforming a canonical set of 3D Gaussians through a positional-encoded MLP of frame time t. Although fitted to a continuous variable, the MLP couples no two values of t in its architecture and effectively predicts discrete per-frame offsets, leaving temporal smoothness to emerge only as a byproduct of optimisation. We redesign the deformation field as a stack of Closed-form Continuous-time (CfC) cells, a Liquid Neural Network (LNN), that is the closed-form solution of the Liquid Time-constant ODE while preserving every other part of the D-3DGS pipeline. Each cell exposes a sigmoidal time gate that interpolates between two candidate hidden states, baking a learned smooth response to t into the loss landscape without invoking any numerical solver. On the eight D-NeRF and seven NeRF-DS scenes the liquid field matches or exceeds the MLP baseline in aggregate, with its largest gains concentrated on the scenes with the most high-frequency articulated motion. The result is a near-zero-friction architectural design that turns the discrete MLP deformation field into an explicit continuous-time function of t.

Abstract

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