Controlled SDEs for Long-Horizon Motion Generation under Latent Decision Uncertainty

Long-horizon motion prediction under external commands is challenged by latent decision uncertainty, where the internal states governing future behavior are unobservable and evolve stochastically over time. This issue is particularly pronounced in biological agents, whose motion trajectories reflect decision-making processes rooted in underlying cognitive states. To address these challenges, we propose CogSDE, a formulation of a controlled stochastic differential equation (SDE) for modeling instruction-driven latent decision dynamics. The drift term in the SDE incorporates a dual-channel control modulation mechanism, enabling external commands to modulate the evolution of latent states. The diffusion term employs a state-dependent operator to model intrinsic uncertainty in latent decision dynamics. Furthermore, we establish dissipativity-based mean-square boundedness for the latent decision dynamics. Experiments demonstrate that CogSDE consistently improves predictive accuracy in long-horizon motion generation. Importantly, predicted trajectories remain well aligned with control commands over extended horizons, a property widely recognized as challenging in long-horizon motion prediction.