DeepAI AI Chat
Log In Sign Up

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

by   Samuel E. Otto, et al.

Data-driven reduced-order models often fail to make accurate forecasts of high-dimensional nonlinear 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 identical 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 an axisymmetric jet flow simulation with 10^5 state variables.


page 1

page 2

page 3

page 4


Balanced Truncation Model Reduction for Lifted Nonlinear Systems

We present a balanced truncation model reduction approach for a class of...

Nonlinear proper orthogonal decomposition for convection-dominated flows

Autoencoder techniques find increasingly common use in reduced order mod...

Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders

Nearly all model-reduction techniques project the governing equations on...

Non-intrusive Nonlinear Model Reduction via Machine Learning Approximations to Low-dimensional Operators

Although projection-based reduced-order models (ROMs) for parameterized ...

Reduced order modeling of nonlinear microstructures through Proper Orthogonal Decomposition

We apply the Proper Orthogonal Decomposition (POD) method for the effici...

Kernel Methods for the Approximation of Nonlinear Systems

We introduce a data-driven order reduction method for nonlinear control ...

Wavelet Adaptive Proper Orthogonal Decomposition for Large Scale Flow Data

The proper orthogonal decomposition (POD) is a powerful classical tool i...