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 the 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 latent state evolution. The diffusion term in the SDE adopts a state-dependent operator to model intrinsic uncertainty in latent decision dynamics. Furthermore, we establish a theoretical upper bound on the long-horizon prediction of CogSDE through dissipativity-based analysis. Experiments demonstrate that CogSDE consistently improves predictive accuracy in long-horizon motion generation. Importantly, predicted trajectories remain well aligned with control instructions over extended horizons, a property widely recognized as challenging in long-horizon motion prediction.