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A CFL-type Condition and Theoretical Insights for Discrete-Time Sparse Full-Order Model Inference

Leonidas Gkimisis, Süleyman Yıldız, Peter Benner, Thomas Richter

TL;DR

This work develops a sparse Full‑Order Model (sFOM) framework for data‑driven, discrete‑time inference with adjacency‑based sparsity. It establishes a theoretical connection between $\ell_2$ regularization and stability via Gershgorin circles, derives a Taylor‑based sampling CFL condition for linear 1D problems, and validates these insights through 1D linear diffusion/advection as well as nonlinear 2D Burgers’ and lid‑driven cavity tests. Key contributions include a closed‑form LS solution, the sampling CFL bound, and practical demonstrations of accurate, stable nonlinear sFOMs under data limitations, along with discussions of stability limitations and potential projection methods to enforce constraints. The results provide actionable guidance on regularization and data‑sampling for stable, interpretable data‑driven discretizations of PDE dynamics, with implications for digital twins and real‑time inference in fluid and transport problems.

Abstract

In this work, we investigate the data-driven inference of a discrete-time dynamical system via a sparse Full-Order Model (sFOM). We first formulate the involved Least Squares (LS) problem and discuss the need for regularization, indicating a connection between the typically employed $l_2$ regularization and the stability of the inferred discrete-time sFOM. We then provide theoretical insights considering the consistency and stability properties of the inferred numerical schemes that form the sFOM and exemplify them via illustrative, 1D test cases of linear diffusion and linear advection. For linear advection, we analytically derive a "sampling CFL" condition, which dictates a bound for the ratio of spatial and temporal discretization steps in the training data that ensures stability of the inferred sFOM. Finally, we investigate the sFOM inference for two nonlinear problems, namely a 2D Burgers' test case and the incompressible flow in an oscillating lid driven cavity, and draw connections between the theoretical findings and the properties of the inferred, nonlinear sFOMs.

A CFL-type Condition and Theoretical Insights for Discrete-Time Sparse Full-Order Model Inference

TL;DR

This work develops a sparse Full‑Order Model (sFOM) framework for data‑driven, discrete‑time inference with adjacency‑based sparsity. It establishes a theoretical connection between regularization and stability via Gershgorin circles, derives a Taylor‑based sampling CFL condition for linear 1D problems, and validates these insights through 1D linear diffusion/advection as well as nonlinear 2D Burgers’ and lid‑driven cavity tests. Key contributions include a closed‑form LS solution, the sampling CFL bound, and practical demonstrations of accurate, stable nonlinear sFOMs under data limitations, along with discussions of stability limitations and potential projection methods to enforce constraints. The results provide actionable guidance on regularization and data‑sampling for stable, interpretable data‑driven discretizations of PDE dynamics, with implications for digital twins and real‑time inference in fluid and transport problems.

Abstract

In this work, we investigate the data-driven inference of a discrete-time dynamical system via a sparse Full-Order Model (sFOM). We first formulate the involved Least Squares (LS) problem and discuss the need for regularization, indicating a connection between the typically employed regularization and the stability of the inferred discrete-time sFOM. We then provide theoretical insights considering the consistency and stability properties of the inferred numerical schemes that form the sFOM and exemplify them via illustrative, 1D test cases of linear diffusion and linear advection. For linear advection, we analytically derive a "sampling CFL" condition, which dictates a bound for the ratio of spatial and temporal discretization steps in the training data that ensures stability of the inferred sFOM. Finally, we investigate the sFOM inference for two nonlinear problems, namely a 2D Burgers' test case and the incompressible flow in an oscillating lid driven cavity, and draw connections between the theoretical findings and the properties of the inferred, nonlinear sFOMs.
Paper Structure (13 sections, 1 theorem, 45 equations, 7 figures)

This paper contains 13 sections, 1 theorem, 45 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Adjacency-based Sparsity
  3. Problem Formulation
  4. Regularization strategies
  5. Theoretical Insights
  6. Closed-form solution
  7. Example: Linear diffusion
  8. A sampling CFL condition for linear, 1D problems
  9. Example: Linear Advection
  10. Nonlinear test cases
  11. 2D Burgers' equation
  12. Lid-Driven Cavity
  13. Conclusion & Future Directions

Key Result

Theorem 1

If the selected $Q_i$ and $\mathbf{f}$ for the inference of $\bm{\beta}_i$ in LSform are the same as those employed for data collection via coefs_disc and if $\mathcal{D}_i$ in LSform has full column rank, there exists a unique solution $\bm{\beta}_i$ to LSform. Furthermore, the solution $\bm{\beta}

Figures (7)

  • Figure 1: Numerical results for the inference of linear diffusion dynamics \ref{['diff_soln']} via \ref{['LSform']} and \ref{['daug']}. Top: Both \ref{['LSform']} and \ref{['daug']} yield accurate results for the state prediction, though training data discretization in space ($\Delta x$) and time ($\Delta t$) affect the accuracy of the augmented data solution from \ref{['daug']}. Bottom: Linear increase in the prediction average error $e$ at $t=10$, for different training data time discretizations $\Delta t$ and $\Delta x=0.24$. For very low $\Delta t$ values, \ref{['daug']} yields unstable results.
  • Figure 2: Schematic representation of sampling CFL condition: If the timestep $\Delta t$ is chosen large enough, the system dynamics cannot be captured by the red measurements in the stencil $Q_i$.
  • Figure 3: Discrete-time sFOM for linear advection: For a given numerical stencil, the stability of the inferred sFOM depends on the discretization of the training data $\Delta x, \Delta t$, following the sampling CFL condition in \ref{['CFL_linadv']}. The 3-pt stencil sFOM prediction for data with $\{\Delta x = 0.008, \; \Delta t = 0.008 \}$ (marked with a black cross) is given on the upper right corner of the figure: The stability condition \ref{['CFL_linadv']} is slightly violated, but the results in finite time remain accurate.
  • Figure 4: sFOM predictions for 2D Burgers' test case, at time $T=10\;s$, for $a=1$ and $\mu=10$: Successful reproduction of the system dynamics in the training data, with the solution of only 1 LS problem, while all eigenvalues of $\mathbf{A}$ lie within the unit circle.
  • Figure 5: Average, relative error of the inferred sFOM \ref{['2Dburg']} at $T=10 \; s$ for different initial conditions: The sFOM model makes accurate predictions for values of $\alpha \leq 1$ and small $\mu$ in \ref{['ini_burg']}. The average, relative error for $\alpha=1$, $\mu=10$ corresponds to the state prediction in \ref{['fig:2Dbrg']}.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof