Stable Nonlinear Dynamical Approximation with Dynamical Sampling

2025/05/17 by Daan Bon, Bon, Daan, Benjamin Caris +3
Computer Science · Decision Sciences · Physics and Astronomy · #32W99 #35C99 #65D15 #65D30 #65D40 #65D99 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2505.11938

openalex publication_date 2025/05/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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Related

• Dynamical Sampling: A Survey
• Residual-Based Time Discretization on Nonlinear Approximation Manifolds: Analysis and Gaussian Applications
• Numerical Method in Classical Dynamics
• Dynamical approximation and sensor placement for filtering problems
• Regularized dynamical parametric approximation