Taming High-Dimensional Dynamics: Learning Optimal Projections onto Spectral Submanifolds

High-dimensional nonlinear systems pose considerable challenges for modeling and control across many domains, from fluid mechanics to advanced robotics. Such systems are typically approximated with reduced-order models, which often rely on orthogonal projections, a simplification that may lead to large prediction errors. In this work, we derive optimality of fiber-aligned projections onto spectral submanifolds, preserving the nonlinear geometric structure and minimizing long-term prediction error. We propose a data-driven procedure to learn these projections from trajectories and demonstrate its effectiveness through a 180-dimensional robotic system. Our reduced-order models achieve up to fivefold improvement in trajectory tracking accuracy under model predictive control compared to the state of the art.