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

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.

FLAG-4D: Flow-Guided Local-Global Dual-Deformation Model for 4D Reconstruction

We introduce FLAG-4D, a novel framework for generating novel views of dynamic scenes by reconstructing how 3D Gaussian primitives evolve through space and time. Existing methods typically rely on a single Multilayer Perceptron (MLP) to model temporal deformations, and they often struggle to capture complex point motions and fine-grained dynamic details consistently over time, especially from sparse input views. Our approach, FLAG-4D, overcomes this by employing a dual-deformation network that dynamically warps a canonical set of 3D Gaussians over time into new positions and anisotropic shapes. This dual-deformation network consists of an Instantaneous Deformation Network (IDN) for modeling fine-grained, local deformations and a Global Motion Network (GMN) for capturing long-range dynamics, refined through mutual learning. To ensure these deformations are both accurate and temporally smooth, FLAG-4D incorporates dense motion features from a pretrained optical flow backbone. We fuse these motion cues from adjacent timeframes and use a deformation-guided attention mechanism to align this flow information with the current state of each evolving 3D Gaussian. Extensive experiments demonstrate that FLAG-4D achieves higher-fidelity and more temporally coherent reconstructions with finer detail preservation than state-of-the-art methods.