# Optimizing Oblique Projections for Nonlinear Systems using Trajectories

arXiv: https://arxiv.org/abs/2106.01211 · HTML: https://mlsanity.com/paper/2106.01211/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jun 2021.

## Built on by (most cited first, 6 of 6)

- [Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoders](https://mlsanity.com/paper/2307.15288/) (2023)
- [Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance](https://mlsanity.com/paper/2207.14387/) (2022)
- [Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds](https://mlsanity.com/paper/2603.15091/) (2026)
- [Toward Adaptive Non-Intrusive Reduced-Order Models: Design and Challenges](https://mlsanity.com/paper/2602.11378/) (2026)
- [Nonlinear Model Order Reduction of Dynamical Systems in Process Engineering: Review and Comparison](https://mlsanity.com/paper/2506.12819/) (2025)
- [Uncovering Extreme Event Mechanisms for Prediction and Control with Sensitivity-Balanced Projections](https://mlsanity.com/paper/2606.05618/) (2026)

## Abstract (arXiv)

Reduced-order modeling techniques, including balanced truncation and H2-optimal model reduction, exploit the structure of linear dynamical systems to produce models that accurately capture the dynamics. For nonlinear systems operating far away from equilibria, on the other hand, current approaches seek low-dimensional representations of the state that often neglect low-energy features that have high dynamical significance. For instance, low-energy features are known to play an important role in fluid dynamics where they can be a driving mechanism for shear-layer instabilities. Neglecting these features leads to models with poor predictive accuracy despite being able to accurately encode and decode states. In order to improve predictive accuracy, we propose to optimize the reduced-order model to fit a collection of coarsely sampled trajectories from the original system. In particular, we optimize over the product of two Grassmann manifolds defining Petrov-Galerkin projections of the full-order governing equations. We compare our approach with existing methods including proper orthogonal decomposition, balanced truncation-based Petrov-Galerkin projection, quadratic-bilinear balanced truncation, and the quadratic-bilinear iterative rational Krylov algorithm. Our approach demonstrates significantly improved accuracy both on a nonlinear toy model and on an incompressible (nonlinear) axisymmetric jet flow with 105 states.