Nonlinear compressive reduced basis approximation : when Taylor meets Kolmogorov

TL;DR

This paper explores advanced nonlinear model reduction techniques for parameter-dependent PDEs, emphasizing the limitations of traditional linear methods and proposing nonlinear, machine learning-based approaches to overcome the Kolmogorov barrier.

Contribution

It provides a theoretical analysis of sensing numbers in nonlinear reduced basis methods and advocates for more expressive nonlinear mappings, including machine learning, to improve approximation efficiency.

Findings

Local sensing number n = p is optimal.

Quadratic mappings are insufficient for wide parameter ranges.

Nonlinear, machine learning-based mappings are necessary for better approximation.

Abstract

This paper investigates model reduction methods for efficiently approximating the solution of parameter-dependent PDEs with a multi-parameter vector $\vec{\mu} \in \mathbb{R}^p$. In cases where the Kolmogorov $N$-width decays fast enough, it is effective to approximate the solution as a sum of $N$ separable terms, each being the product of a parameter-dependent coefficient and a space-dependent function. This leads to reduced-order models with $N$ degrees of freedom and complexity of order ${\mathcal O}(N^3)$. However, when the $N$-width decays slowly, $N$ must be large to achieve acceptable accuracy, making cubic complexity prohibitive. The linear complexity measure in terms of Kolmogorov width must be replaced by the Gelfand width, with its associated sensing number. Recent nonlinear approaches based on this notion decompose the $N$ coordinates into two groups: $n$ free variables and $\overline{n}$ dependent variables, where the latter are nonlinear functions of the former ($N= n+\overline n$). Several works have focused on cases where these $\overline{n}$ functions are homogeneous quadratic forms of the $n$ variables, with optimization strategies for choosing $n$ given a target accuracy. A rigorous analysis of the local sensing number is carried out, showing that $n = p$ is optimal and appropriate, at least locally, around a reference point. In practical scenarios involving wide parameter ranges, the condition $p\le n \le p + k$ (with $k$ small) is valid and more robust from continuity arguments. Additionally, the assumption of a quadratic mapping, while justified in a local sense, becomes insufficient. More expressive nonlinear mappings-including those using machine learning-become necessary. This work contributes a theoretical foundation for such strategies and highlights the need for further investigations to push back the Kolmogorov Barrier.