Neural Autonomous Differential Equations: Integration-Free Learning for Time-Invariant Dynamics

We consider the problem of learning autonomous dynamics from observed trajectories. Neural ODEs, which model the vector field with a neural network, have emerged as a popular framework for this task. These methods backpropagate through adaptive ODE solvers, which makes them computationally expensive and numerically sensitive. Recently proposed variants accelerate training, but either weaken the ability to constrain autonomous dynamics, introduce periodicity assumptions, or require trajectory truncation. We propose Neural Autonomous Differential Equations (NADE), a simple solver-free training method. NADE separates trajectory reconstruction from vector-field estimation by (i) fitting the trajectory using differentiable splines, and (ii) training the velocity field using analytic derivatives of the fitted trajectory at sampled states. At inference time, the learned autonomous field can be integrated with any standard numerical solver. This preserves the inductive bias of autonomous dynamics while avoiding solver-based training. We evaluate NADE on multistable, periodic, and chaotic systems. Under equal wall-clock training budgets, it achieves substantially faster convergence and lower training error than standard Neural ODEs and accelerated variants.