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 ϕ([u;h])→z feeds four heads: g,hcand form two candidate states, while Waz,Wbz form the affine elapsed-time term. The time gate is the only path through which τ enters the cell; we add a Gaussian diffusion increment λε to its pre-activation ( στ=σ(Wazτ+Wbz+λε) ), 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 ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
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 (↑) , SSIM (↑) and LPIPS (↓) at 800×800 test images. D-NeRF and TiNeuVox numbers are quoted from [ 17 ] ; D-3DGS (MLP) and Ours (SDE-LNN, λ=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 ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
PSNR ↑
SSIM ↑
LPIPS ↓
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, λ=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)
λ
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 λ . λ=0 is the deterministic CfC. best per column.
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.
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.
Mingzhao Li, Arghya Pal, Guan Yuan Tan
School of Information Technology, Monash University, Selangor, Malaysia
Dynamic 3D Gaussian Splatting methods achieve strong training-view PSNR on monocular video but generalize poorly: on the D-NeRF benchmark we measure an average train-test PSNR gap of 6.18 dB, rising to 11 dB on individual scenes. We report two findings that together account for most of that gap. Finding 1 (the role of splitting). A systematic ablation of the Adaptive Density Control pipeline (split, clone, prune, frequency, threshold, schedule) shows that splitting is responsible for over 80% of the gap: disabling split collapses the cloud from 44K to 3K Gaussians and the gap from 6.18 dB to 1.15 dB. Across all threshold-varying ablations, gap is log-linear in count (r = 0.995, bootstrap 95% CI [0.99, 1.00]), which suggests a capacity-based explanation. Finding 2 (the role of deformation coherence). We show that the capacity explanation is incomplete. A local-smoothness penalty on the per-Gaussian deformation field -- Elastic Energy Regularization (EER) -- reduces the gap by 40.8% while growing the cloud by 85%. Measuring per-Gaussian strain directly on trained checkpoints, EER reduces mean strain by 99.72% (median 99.80%) across all 8 scenes; on 8/8 scenes the median Gaussian under EER is less strained than the 1st-percentile (best-behaved) Gaussian under baseline. Alongside EER, we evaluate two further regularizers: GAD, a loss-rate-aware densification threshold, and PTDrop, a jitter-weighted Gaussian dropout. GAD+EER reduces the gap by 48%; adding PTDrop and a soft growth cap reaches 57%. We confirm that coherence generalizes to (a) a different deformation architecture (Deformable-3DGS, +40.6% gap reduction at re-tuned lambda), and (b) real monocular video (4 HyperNeRF scenes, reducing the mean PSNR gap by 14.9% at the same lambda as D-NeRF, with near-zero quality cost). The overfitting in dynamic 3DGS is driven by incoherent deformation, not parameter count.
We introduce a probabilistic splat-based radiance field framework that retains the fast rasterization and test-time efficiency of 3D Gaussian Splatting (3DGS) while replacing heuristic primitive manipulation with gradient-based optimization of a volumetric probability density. Rather than relocating, splitting, or culling Gaussians via hand-tuned densification (e.g., ADC), we treat primitive locations as samples drawn from a persistent, learnable density. We instantiate this density using a novel, memory-efficient multi-scale hierarchical grid that enables end-to-end gradient-based optimization. To stabilize the optimization, we derive an unbiased gradient estimator with control variates that markedly reduces variance. By allowing probability mass to flow to where the loss demands, our framework eliminates brittle priors and naturally explores the volume, achieving state-of-the-art reconstruction quality on mip-NeRF 360 while preserving 3DGS-level rendering speed.
Mia Gaia Polansky, George Kopanas, Stephan Garbin +2