Nicola Farenga

Learning the continuous-time evolution of nonlinear dynamical systems usually relies on trajectory objectives that simulate the learned dynamics during training and backpropagate through the solver, at a cost growing with the rollout horizon. Velocity matching instead regresses the velocity field directly, using piecewise-linear interpolation of observed trajectories and points sampled uniformly within each interval, so no simulation is needed. On nonlinear ODEs and semi-discretized time-dependent PDEs it attains errors more than an order of magnitude lower than trajectory-based objectives under a common iteration budget, and keeps accuracy under schemes and step sizes not used in training.

Learning the continuous-time evolution of nonlinear dynamical systems usually relies on trajectory objectives that simulate the learned dynamics during training and backpropagate through the solver, at a cost growing with the rollout horizon. Velocity matching instead regresses the velocity field directly, using piecewise-linear interpolation of observed trajectories and points sampled uniformly within each interval, so no simulation is needed. On nonlinear ODEs and semi-discretized time-dependent PDEs it attains errors more than an order of magnitude lower than trajectory-based objectives under a common iteration budget, and keeps accuracy under schemes and step sizes not used in training.