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Stable Nonlinear Dynamical Approximation with Dynamical Sampling

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arxiv 2505.11938 v1 pith:XB4EQOS3 submitted 2025-05-17 math.NA cs.NA

Daan Bon , Benjamin Caris , Olga Mula This is my paper · ORCID
classification math.NAcs.NA
keywords dynamicalnonlinearapproximationfunctionsmethodprojectionssamplingstability
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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.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A Dynamical Approximation Scheme on the Stiefel manifold for Wasserstein Gradient Flows

    math.NA 2026-07 conditional novelty 6.0 of 10

    A Stiefel-manifold dynamical approximation for Wasserstein gradient flows represents the evolving transport map by a moving linear subspace and controls the Wasserstein error through an adaptive background space.

  2. Residual-Based Time Discretization on Nonlinear Approximation Manifolds: Analysis and Gaussian Applications

    math.NA 2026-07 conditional novelty 6.0 of 10

    Residual-minimization time steppers on nonlinear manifolds obey error bounds of order h or h² plus a residual term, with an extra conditioning term for the Dirac-Frenkel-based variant.

  3. Dirac-Frenkel dynamics with inertia for nonlinearly parametrized solutions of evolution problems

    math.NA 2026-06 unverdicted novelty 6.0 of 10

    Inertial Dirac-Frenkel dynamics yield well-posed parameter evolution for nonlinear parametrizations with a posteriori error bounds and improved numerical robustness.

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