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 computationally tractable procedure to approximate these projections from data, and show how the effect of control can be incorporated. For a 180-dimensional robotic system, we demonstrate that our reduced-order models outperform previous state-of-the-art approaches by up to fivefold in trajectory tracking accuracy under model predictive control.