Stable Nonlinear Dynamical Approximation with Dynamical Sampling

TL;DR

This paper introduces a stable nonlinear dynamical approximation method for time-dependent PDEs using parametrized decoder functions and a novel dynamical sampling strategy with stability guarantees.

Contribution

It presents a new approach combining parametrized decoders and a sampling strategy for stable, accurate PDE approximations with low computational complexity.

Findings

Effective on several moderate-dimensional PDE examples

Provides stability guarantees for the sampling method

Achieves accurate approximations with low numerical complexity

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.