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Exact operator inference with minimal data

Henrik Rosenberger, Benjamin Sanderse, Giovanni Stabile

TL;DR

Numerical results for differential equations involving 2nd, 3rd and 8th order polynomials demonstrate that the novel snapshot data generation method leads to exact reconstruction of the intrusive reduced order models.

Abstract

This work introduces a novel method to generate snapshot data for operator inference that guarantees the exact reconstruction of intrusive projection-based reduced-order models (ROMs). To ensure exact reconstruction, the operator inference least squares matrix must have full rank, without regularization. Existing works have achieved this full rank using heuristic strategies to generate snapshot data and a-posteriori checks on full rank, but without a guarantee of success. Our novel snapshot data generation method provides this guarantee thanks to two key ingredients: first we identify ROM states that induce full rank, then we generate snapshots corresponding to exactly these states by simulating multiple trajectories for only a single time step. This way, the number of required snapshots is minimal and orders of magnitude lower than typically reported with existing methods. The method avoids non-Markovian terms and does not require re-projection. Since the number of snapshots is minimal, the least squares problem simplifies to a linear system that is numerically more stable. In addition, because the inferred operators are exact, properties of the intrusive ROM operators such as symmetry or skew-symmetry are preserved. Numerical results for differential equations involving 2nd, 3rd and 8th order polynomials demonstrate that the novel snapshot data generation method leads to exact reconstruction of the intrusive reduced order models.

Exact operator inference with minimal data

TL;DR

Numerical results for differential equations involving 2nd, 3rd and 8th order polynomials demonstrate that the novel snapshot data generation method leads to exact reconstruction of the intrusive reduced order models.

Abstract

This work introduces a novel method to generate snapshot data for operator inference that guarantees the exact reconstruction of intrusive projection-based reduced-order models (ROMs). To ensure exact reconstruction, the operator inference least squares matrix must have full rank, without regularization. Existing works have achieved this full rank using heuristic strategies to generate snapshot data and a-posteriori checks on full rank, but without a guarantee of success. Our novel snapshot data generation method provides this guarantee thanks to two key ingredients: first we identify ROM states that induce full rank, then we generate snapshots corresponding to exactly these states by simulating multiple trajectories for only a single time step. This way, the number of required snapshots is minimal and orders of magnitude lower than typically reported with existing methods. The method avoids non-Markovian terms and does not require re-projection. Since the number of snapshots is minimal, the least squares problem simplifies to a linear system that is numerically more stable. In addition, because the inferred operators are exact, properties of the intrusive ROM operators such as symmetry or skew-symmetry are preserved. Numerical results for differential equations involving 2nd, 3rd and 8th order polynomials demonstrate that the novel snapshot data generation method leads to exact reconstruction of the intrusive reduced order models.

Paper Structure

This paper contains 26 sections, 9 theorems, 67 equations, 7 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dynamical systems with polynomial terms
  4. Intrusive projection-based model reduction
  5. Operator inference
  6. Insufficiently informative snapshot data
  7. Exact operator inference
  8. Rank-ensuring ROM states and input signals
  9. Snapshot data generation via single-step ensemble
  10. Exact reconstruction of the intrusive ROM operators
  11. Assumptions on the FOM solver
  12. Remarks on changing ROM dimension
  13. Choice of time step size $\Delta t$
  14. Numerical experiments
  15. Chafee-Infante equation / Allen-Cahn equation
  16. ...and 11 more sections

Key Result

Theorem 3.1

For any $n\in\mathbb{N}^+$, ${N_u}\in\mathbb{N}_0$ and $\mathcal{I}\subset\mathbb{N}_0$, the matrix ${\mathbf{P}} := \left[ {\bm{p}}(\breve {\bm{x}}_0,{\bm{u}}_0) \;\;\dots\;\; {\bm{p}}(\breve {\bm{x}}_{K-1},{\bm{u}}_{K-1}) \right] \; \in\mathbb{R}^{{n_f}\times K}$ defined in eq:def p matrix and dot

Figures (7)

  • Figure 1: Relative operator errors $\frac{\|\hat{{\mathbf{O}}} - \tilde{{\mathbf{O}}}\|_F}{\|\tilde{{\mathbf{O}}}\|_F}$.
  • Figure 1: Relative average ROM state error \ref{['eq: rel avg rom state error']} for Chafee-Infante equation.
  • Figure 2: Condition numbers of data matrices ${\mathbf{P}}$.
  • Figure 3: Violation of the energy-preserving constraint of the convection operator of Burgers' equation.
  • Figure 3: Relative average ROM state error \ref{['eq: rel avg rom state error']} for Burgers' equation.
  • ...and 2 more figures

Theorems & Definitions (19)

  • Example 1
  • Example 2
  • Example 3
  • Theorem 3.1: Full rank of matrix ${\mathbf{P}}$
  • Theorem 3.2: Exact reconstruction of intrusive ROM operators
  • Proof 1
  • Theorem A.1: Gappy multivariate polynomial interpolation
  • Theorem A.2: Multivariate polynomial interpolation without gaps
  • Theorem A.3: Homogeneous multivariate polynomial interpolation
  • Proof 2: Proof of Theorem \ref{['theo:homogen poly']}
  • ...and 9 more