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arXiv · 2505.11938

Stable Nonlinear Dynamical Approximation with Dynamical Sampling

Daan Bon·Benjamin Caris·Olga Mula
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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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Daan Bon, Benjamin Caris, Olga Mula. 2025-05-17. Stable Nonlinear Dynamical Approximation with Dynamical Sampling. https://arxiv.org/abs/2505.11938

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