The core idea of PT-Flow is to capture the underlying dynamics as a continuous velocity field within a proper latent space. Let \( s_t \in \mathbb{R}^{d_s} \) denote the state input at physical time \( t \), and \( z_t \in \mathbb{R}^{d_z} \) be the latent representation of \( s_t \). Here, \( t \) represents the physical time elapsed after starting at the initial state \( s_0 \). The dynamics are then modeled by learning a time-dependent velocity field via an ODE formulation.

To enable such continuous-time modeling, PT-Flow features two key designs:

We instantiate the PT-Flow paradigm as ODEWorld, a highly versatile and efficient continuous-time predictive architecture for both high-fidelity video generation and effective robotic policy learning. Apart from building PT-Flow, we first employ a frozen DINOv2 encoder as our vision backbone \( f_{\text{obs}} \) and a dedicated image decoder \( g_{\text{obs}} \) to reconstruct DINO features back to images \( \hat{x}_t \) for video generation. We further embed PT-Flow within the DINO feature space.

In our experiments, we find that a single token is sufficient to model \( z_t \) while achieving strong performance. Specifically, \( z_t \in \mathbb{R}^{1 \times 768} \) is highly compact compared to \( s_t \in \mathbb{R}^{16 \times 16 \times 768} \) in the original DINO space. More importantly, this result further supports the key insight of latent world modeling: raw pixels contain substantial redundancy, while the underlying dynamics can be effectively captured by a much smaller latent representation.

Latent space analysis.We compare the latent space of ODEWorld against the raw DINO feature space and the latent space of a discrete next-step predictive baseline. PCA visualizations of latent trajectories across multiple LIBERO tasks show that ODEWorld produces the smoothest latent trajectories. Crucially, while improving dynamical smoothness, ODEWorld preserves a manifold topology remarkably similar to that of the original DINO feature space.

Ablation on Dynamical Decoupling. We observe that Decoupling is critical for continuous dynamics modeling. Without separating static context from the state space, the learned latent trajectories become noisy and unstable. Conditioning on \( s_0 \) enables the latent representation to focus on dynamics, leading to smooth and meaningful latent flows.

Evaluation of Quality and Efficiency. ODEWorld consistently outperforms baselines in both pixel-level and perceptual metrics across short- and long-term horizons, while achieving higher inference speed and a smaller model size.

Comparison of Generated Samples. ODEWorld enables long-horizon video generation, producing temporally coherent and physically plausible videos that closely match the ground-truth observations. In contrast, baseline methods often suffer from blurry visual predictions and hallucinations in long-horizon generation.

ODEWorld for Policy Learning. We assess how ODEWorld can empower downstream policy learning by inferring latent subgoals as guidance. We instantiate three guidance paradigms: (1) velocity-conditioned policy \( \pi(a \mid z, c, v) \), (2) single-subgoal policy \( \pi(a \mid z, c, \hat{z}_{\tau}) \), where \( \tau = 0.25 \), (3) sequential-subgoal policy \( \pi(a \mid z, c, \{\hat{z}_{\tau_i}\}_{i=1}^{n}) \), where \( n = 5 \), \( \tau_i = 0.05 i \), where subgoals are generated via ODE integration. \( c \) denotes image goal and language instruction. Results show that all ODEWorld variants outperform baselines on LIBERO-LONG tasks, with the sequential-subgoal paradigm achieving the highest success rate. This performance gain demonstrates the high-fidelity temporal consistency of ODEWorld rollouts. and such consistent sequential subgoal guidance effectively benefits downstream policy learning.