Dissipative Neural SDEs for Robust Dynamical Learning

While Neural stochastic differential equations (SDEs) provide a flexible framework for learning stochastic dynamics, their training still lacks dynamical guarantees, and in particular, they often exhibit non-physical behaviors under distribution shifts. This work aims to mitigate such failures by emphasizing the role of the Foster-Lyapunov dissipativity condition, which is commonly assumed in the current literature but is not guaranteed by the standard Neural SDE architectures. In this work, we introduce a flexible unified framework of partially dissipative Neural SDEs, which accommodates a wide class of dynamics, such as globally stable, chaotic, and integrated dynamics. We translate the dissipativity condition into sufficient forms compatible with $2$ enforcement mechanisms: PINN-style regularization and structural projection. Theoretically, we show that structural projection is conditionally no-regret, revealing a stability–flexibility trade-off, while our analysis further supports the framework's pathwise approximation flexibility over a moment-bounded class of initial laws. Experimentally, we demonstrate that the methods considerably improve dynamical recovery and robustness. The practical relevance is demonstrated on real-world high-dimensional datasets in terms of forecasting and reconstruction performance under context missingness and noise corruption.