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Learning physics-based reduced-order models from data using nonlinear manifolds

Rudy Geelen, Laura Balzano, Stephen Wright, Karen Willcox

Abstract

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

Learning physics-based reduced-order models from data using nonlinear manifolds

Abstract

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.
Paper Structure (14 sections, 24 equations, 19 figures, 2 tables, 4 algorithms)

This paper contains 14 sections, 24 equations, 19 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction and background
  2. Learning nonlinear manifolds
  3. A general representation learning problem
  4. Computing the basis expansion
  5. Orthogonal subspace transformations: an illustrative example
  6. Learning reduced-order models on nonlinear manifolds
  7. Projection-based model reduction
  8. Learning physics-based reduced-order models from data
  9. Numerical experiments
  10. Practical considerations
  11. The Allen-Cahn equation
  12. The Korteweg-de Vries equation
  13. Incompressible Navier-Stokes -- Flow past a cylinder
  14. Conclusion & Discussion

Figures (19)

  • Figure 1: Comparison of (a) the linear-subspace POD with (b) the POD-based representation learning and (c) alternating minimization based representation learning approaches for reconstructing, using two modal coefficients, the three-dimensional manifold. The relative state error $\| \mathbf{S}-\boldsymbol{\Gamma}(\hat{\mathbf{S}}) \|_F/ \| \mathbf{S} - \mathbf{S}_\text{ref} \|_F$ is given between parentheses. The gray surface denotes the original three-dimensional manifold, whereas the colored surfaces illustrate the different reconstructions. The black arrows represent the basis vectors.
  • Figure 2: Normalized singular values of the centered snapshot matrix for the Allen-Cahn problem. The blue and red areas denote the singular values whose corresponding left singular vectors are columns in $\mathbf{V}$ and $\overline{\mathbf{V}}$, respectively, in the MPOD-OpInf formulation.
  • Figure 3: Comparison of the reference solution at parameter value $\mu=0.5127$ (top) with the reconstructions from the two-equation OpInf (middle) and MPOD-OpInf (bottom) models for the Allen-Cahn model.
  • Figure 4: Pointwise error in the reconstructions for the test trajectory at parameter value $\mu=0.5127$ for the Allen-Cahn model.
  • Figure 5: Normalized singular values of the mean-subtracted snapshot matrix for the Korteweg-de Vries problem. The blue and red areas denote the singular values whose corresponding left singular vectors are columns in $\mathbf{V}$ and $\overline{\mathbf{V}}$, respectively, in the MPOD-OpInf formulation.
  • ...and 14 more figures

Theorems & Definitions (1)

  • Example 3.1