No more step-by-step simulation in training: velocity matching cuts trajectory error by an order of magnitude and survives new step sizes

Training a continuous-time dynamical system no longer requires unrolling the learned dynamics; the velocity field can be regressed directly at a training cost comparable to one-step teacher forcing. Trajectory-based objectives numerically integrate the learned system during training and backpropagate through the integration steps, with cost growing with the rollout horizon; one-step teacher forcing avoids simulation but approximates the velocity objective only to first order. On the Euler equations of a free rigid body, the Stuart–Landau oscillator and an active particle in the Müller–Brown potential, velocity matching reduces the test trajectory error by up to an order of magnitude relative to rollout strategies under a common iteration budget, while the latter yield test errors of 10^-1 or larger.

Velocity matching constrains the learned velocity field along the piecewise-linear interpolant of observed trajectories and supervises it at points sampled uniformly within each time interval. Teacher forcing supervises the velocity field only at observation times, with a first-order finite difference as the velocity target; velocity matching samples along the interpolant chord and uses the same finite difference as the target. Each step evaluates the learned field at a single collocation point per sample, so the per-iteration cost matches an explicit Euler teacher-forcing update, and the batch size controls the variance of the gradient estimate.

The stochastic sampling yields two properties: a second-order approximation of the velocity population risk and an implicit matching of the learned Jacobian's action along observed trajectories. Teacher forcing approximates the velocity population risk only to first order; Jacobian matching previously required access to the true Jacobian, at a cost of derivative evaluations and memory per training state. Across the three ODE test cases, the test Jacobian error of velocity matching is lower than that of trajectory-based strategies by a factor of 3 to 9, although no Jacobian information is used during training.

The learned velocity field can be simulated at inference with numerical schemes and step sizes different from those used in training without loss of accuracy. Updates learned by trajectory-based strategies are accurate only with the training scheme and step size; refining the step size increases their error, which saturates at a higher plateau. On the Euler, Stuart–Landau and active Müller–Brown systems, the velocity-matching error coincides at large step sizes with the error of integrating the true velocity field with the same scheme, and as the step size shrinks it saturates at a plateau no higher than the training-discretization error.

The demonstrated scope is nonlinear ODEs and one-dimensional periodic-domain semi-discretized time-dependent PDEs under noise-free observations, with training, validation and test sharing one architecture and optimization hyperparameters, and a test horizon twice the training horizon. Continuous-time approximation is assessed on the ODE benchmarks, where the true velocity field is available in closed form.

The constants in the stability estimates grow exponentially with the time horizon and the Lipschitz constant of the learned velocity field, and may become uninformative. The effect of stochastic collocation on noisy observations and on specific classes of dynamics, whether oscillatory, conservative, dissipative or multiscale, remains largely unexplored, and how to design collocation distributions adapted to system classes is an open question.