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Stable Nonlinear Dynamical Approximation with Dynamical Sampling

Daan Bon, Benjamin Caris, Olga Mula

TL;DR

This work proposes a dynamical sampling strategy which comes with stability guarantees, while keeping a low numerical complexity, and shows the effectiveness of the method on several examples in moderate spatial dimension.

Abstract

We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and analyzing stability and accuracy of nonlinear dynamical approximations. The parameters of these functions are evolved in time by means of projections on finite dimensional subspaces of an ambient Hilbert space related to the PDE evolution. For practical computations of these projections, one usually needs to sample. We propose a dynamical sampling strategy which comes with stability guarantees, while keeping a low numerical complexity. We show the effectiveness of the method on several examples in moderate spatial dimension.

Stable Nonlinear Dynamical Approximation with Dynamical Sampling

TL;DR

This work proposes a dynamical sampling strategy which comes with stability guarantees, while keeping a low numerical complexity, and shows the effectiveness of the method on several examples in moderate spatial dimension.

Abstract

We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and analyzing stability and accuracy of nonlinear dynamical approximations. The parameters of these functions are evolved in time by means of projections on finite dimensional subspaces of an ambient Hilbert space related to the PDE evolution. For practical computations of these projections, one usually needs to sample. We propose a dynamical sampling strategy which comes with stability guarantees, while keeping a low numerical complexity. We show the effectiveness of the method on several examples in moderate spatial dimension.
Paper Structure (25 sections, 105 equations, 16 figures)

This paper contains 25 sections, 105 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Nonlinear Dynamical Approximation
  3. Approximation classes generated by decoders
  4. Dynamical Approximation Scheme: Abstract Framework
  5. A guiding class of PDEs: gradient flows of lambda-convex functionals
  6. The setting Theta=Rn and V=L2 and challenges for numerical implementation
  7. Dynamical Approximation with linear observations
  8. Formulation
  9. Example:
  10. Complexity:
  11. Refined Formulation:
  12. Error Analysis
  13. The case of gradient flows with a convex potential
  14. Inexact Observations
  15. Dynamical Sampling
  16. ...and 10 more sections

Figures (16)

  • Figure 1: The degrees of freedom of the nonlinear decoder evolve in time to preserve stability and near-optimality pointwise in time (\ref{['sec:nonlinear dynamical approximation']} and \ref{['sec:dyn-approx-obs']}). Around the black dots, local information is collected to best approximate the solution, and the dots move together with the numerical solution thanks to a data-driven strategy (\ref{['sec:dyn-sampling']}). The images show our dynamical approximation of a Fokker-Planck equation in 2D where the exact solution consists of a mixture of 3 Gaussians (further details in \ref{['sec:FP_results']}).
  • Figure 2: KdV 1D: Exact solution and its approximation ($n=30,\, m=40,\, \sigma=0.1$).
  • Figure 3: KdV 1D: Behaviour of the solution under a varying number of measurements $m$ ($\sigma=0.1$).
  • Figure 4: AC 1D: Exact solution and its approximation ($n=m=45$, $\sigma=0.1$).
  • Figure 5: AC 1D: Behaviour of the solution under a varying number of measurements $m$ ($\sigma=0.1$).
  • ...and 11 more figures

Theorems & Definitions (10)

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  • Remark 5.1
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