Velocity Matching Objective Improves Dynamical System Learning
Original title: Learning to evolve by matching velocities
Simulation-free velocity matching approximates continuous-time evolution without rollout simulation or Jacobian constraints.
Trajectory-based objectives for learning continuous-time nonlinear dynamical systems suffer degeneration as rollout horizons grow and introduce bias from numerical schemes constraining the Jacobian spectrum. The new Velocity Matching objective uses piecewise-linear interpolation and stochastic collocation to approximate velocity population risk to second order without simulation. It connects to A-stable Runge-Kutta methods and matches true Jacobian action along trajectories, yielding consistent minimizers and improved accuracy on ODEs and PDEs while avoiding long-rollout instabilities.
- Simulation-free objective approximates velocity risk to second order
- Connects to A-stable Runge-Kutta schemes
- Matches true Jacobian action along trajectories
- Improves accuracy over trajectory-based methods
- Eliminates rollout simulation costs
From the original article
1 min read at huggingface.co
Learning the continuous-time evolution of nonlinear dynamical systems relies on trajectory-based objectives, which require simulating the learned dynamics during training and backpropagating through the numerical scheme, at a cost that grows with the rollout horizon.
By Nicola Farenga, Andrea Manzoni · huggingface.co
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