# Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariance

arXiv: https://arxiv.org/abs/2207.14387 · HTML: https://mlsanity.com/paper/2207.14387/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 75 references that are in the map.
- First posted on arXiv 28 Jul 2022.

## Builds on (most cited first, 5 of 5)

- [Certified dimension reduction in nonlinear Bayesian inverse problems](https://mlsanity.com/paper/1807.03712/) (2018)
- [Optimizing Oblique Projections for Nonlinear Systems using Trajectories](https://mlsanity.com/paper/2106.01211/) (2021)
- [Balanced Truncation Model Order Reduction For Quadratic-Bilinear Control Systems](https://mlsanity.com/paper/1705.00160/) (2017)
- [Empirical Differential Gramians for Nonlinear Model Reduction](https://mlsanity.com/paper/1902.09836/) (2019)
- [Fast singular value decay for Lyapunov solutions with nonnormal coefficients](https://mlsanity.com/paper/1410.8741/) (2014)

## 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)
- [Operator learning without the adjoint](https://mlsanity.com/paper/2401.17739/) (2024)
- [Operator learning for hyperbolic partial differential equations](https://mlsanity.com/paper/2312.17489/) (2023)
- [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)
- [Uncovering Extreme Event Mechanisms for Prediction and Control with Sensitivity-Balanced Projections](https://mlsanity.com/paper/2606.05618/) (2026)

## Abstract (arXiv)

Data-driven reduced-order models often fail to make accurate forecasts of high-dimensional nonlinear dynamical systems that are sensitive along coordinates with low-variance because such coordinates are often truncated, e.g., by proper orthogonal decomposition, kernel principal component analysis, and autoencoders. Such systems are encountered frequently in shear-dominated fluid flows where non-normality plays a significant role in the growth of disturbances. In order to address these issues, we employ ideas from active subspaces to find low-dimensional systems of coordinates for model reduction that balance adjoint-based information about the system's sensitivity with the variance of states along trajectories. The resulting method, which we refer to as covariance balancing reduction using adjoint snapshots (CoBRAS), is analogous to balanced truncation with state and adjoint-based gradient covariance matrices replacing the system Gramians and obeying the same key transformation laws. Here, the extracted coordinates are associated with an oblique projection that can be used to construct Petrov-Galerkin reduced-order models. We provide an efficient snapshot-based computational method analogous to balanced proper orthogonal decomposition. This also leads to the observation that the reduced coordinates can be computed relying on inner products of state and gradient samples alone, allowing us to find rich nonlinear coordinates by replacing the inner product with a kernel function. In these coordinates, reduced-order models can be learned using regression. We demonstrate these techniques and compare to a variety of other methods on a simple, yet challenging three-dimensional system and a nonlinear axisymmetric jet flow simulation with 105 state variables.