Markovian Dynamics Enforcer: Feasibility Preserving Correction on Learned Dynamics Manifolds

Neural differential equations offer a powerful approach for learning dynamical systems from data. However, they do not inherently respect known constraints, such as conservation laws, that should be obeyed by the learned dynamics. It is well known that enforcing constraints in data-driven models can enhance their generalizability and numerical stability. In this paper, we introduce projected neural differential equations (PNDEs), a method for constraining neural differential equations based on projection of the predicted velocities onto the tangent space of the manifold that fulfills the constraint. In tests on several examples from different fields, including chaotic dynamical systems and power grid models, PNDEs outperform existing methods for constraining learned dynamics, require fewer hyperparameters, and are computationally more efficient. The proposed approach demonstrates potential for enhancing the modeling of constrained dynamical systems, particularly in domains where accuracy and reliability are essential.

Accurate dynamics models are essential for model-based robotic control, yet nominal Euler--Lagrange models often become inaccurate in the presence of payload variation, unmodeled coupling, friction, aerodynamic effects, and changing operating conditions. Most learning-based correction methods improve prediction accuracy by introducing a single additive residual, but do not preserve the internal mechanical structure of Euler--Lagrange systems. This leads to models that do not preserve symmetry, positive-definiteness, or the coupling between inertia and velocity-dependent terms, which can result in physically inconsistent predictions and reduced reliability when embedded in model-based controllers. We propose a structure-preserving residual learning framework that decomposes model mismatch into an inertia correction, the corresponding induced Coriolis term, and a generalized-force residual. The mechanical component is learned under physical constraints, while the disturbance-sensitive component is represented through a sparse history-dependent latent interaction model and adapted online using Bayesian linear regression. This separation preserves key mechanical structure while restricting adaptation to the part of the dynamics most affected by changing conditions. Experiments across multiple robotic platforms, including mobile, aerial, and manipulator systems, show that the proposed method improves dynamics prediction and trajectory tracking under coupled and time-varying dynamics. These results highlight the value of combining structured residual modeling, compact latent interaction selection, and selective online adaptation for real-world model-based control.

Accurate modeling of nonlinear vehicle dynamics is essential for high-speed autonomous racing, where controllers operate at the handling limits. Model-based methods are interpretable but rely on simplifying assumptions, while purely learned models capture nonlinearities yet often lack physical consistency, generalization, and adaptability to changing operating conditions. This paper presents DynSSM, a physics-aware state-space memory framework that combines learned temporal representations with a structured vehicle dynamics model. The proposed approach integrates state-space sequence modeling and recurrent encoders to capture long- and short-term dynamic behavior. It simultaneously adapts tire and vehicle-dynamics parameters within bounded ranges to preserve physical plausibility. A residual correction mechanism compensates for remaining unmodeled dynamics while preserving the underlying physics-based structure. DynSSM is evaluated on both simulated small-scale racing data and real-world full-scale autonomous Indy racecar data. When evaluated on an unseen real-world track, DynSSM reduces one-step prediction RMSE by up to $27.2\%$ in longitudinal velocity, $73.9\%$ in lateral velocity, and $88.8\%$ in yaw rate compared with the state-of-the-art (\sota{}) baselines. Component-wise ablation studies demonstrate the importance of temporal memory, parameter adaptation, and residual correction for predictive performance and robustness. Further, closed-loop simulations using nonlinear model predictive control demonstrate that DynSSM remains feasible across the evaluated tracks and achieves up to a 13.2% reduction in one-lap completion time compared with \sota{} baselines. These results indicate that combining temporal memory, bounded physics-guided parameter adaptation, and residual correction provides an accurate, interpretable, and control-ready dynamics model for autonomous racing.