Tree-Based Nonlinear Reduced Modeling

TL;DR

This work addresses reduced modeling for parametric PDEs by moving beyond single linear subspaces to a tree-structured library of nonlinear approximation spaces. It develops two main strategies: (i) Cartesian dyadic splits of the parameter domain to form a hierarchical library, and (ii) a general tree construction that does not rely on the shape of the parameter space, both guided by greedy selection. A key novelty is extending the framework to general metric spaces, notably the $L^2$-Wasserstein space, via Wasserstein barycenters and a barycentric greedy algorithm. Numerical experiments across diffusion, convection-diffusion, and KdV-type problems show that tree-based libraries can handle weak coercivity, transport-dominated regimes, and general geometry of the parameter domain, offering a meaningful trade-off between online cost and offline offline complexity. The results indicate that the proposed nonlinear, tree-structured library approaches provide broader applicability and competitive performance compared to classical linear reduced models, with several directions left for theoretical analysis and practical implementation of mapping strategies between parameters and local spaces.

Abstract

This paper is concerned with model order reduction of parametric Partial Differential Equations (PDEs) using tree-based library approximations. Classical approaches are formulated for PDEs on Hilbert spaces and involve one single linear space to approximate the set of PDE solutions. Here, we develop reduced models relying on a collection of linear or nonlinear approximation spaces called a library, and which can also be formulated on general metric spaces. To build the spaces of the library, we rely on greedy algorithms involving different splitting strategies which lead to a hierarchical tree-based representation. We illustrate through numerical examples that the proposed strategies have a much wider range of applicability in terms of the parametric PDEs that can successfully be addressed. While the classical approach is very efficient for elliptic problems with strong coercivity, we show that the tree-based library approaches can deal with diffusion problems with weak coercivity, convection-diffusion problems, and with transport-dominated PDEs posed on general metric spaces such as the $L^2$-Wasserstein space.

Figures (14)

• Figure 1: Diffusion problems: Diffusion coefficient $a(x,y)$ for the following values of the parameter $y$: $(0,0)$, $(-1,-1)$, $(0,-1)$, $(0.5,0.5)$, $(1,1)$.
• Figure 2: Diff-1 problem: Comparison of the algorithms $\texttt{plain-greedy}$ and $\mathcal{M}\texttt{-based-split}$ with $\alpha=1$ (left) and $\alpha=0.105$ (right).
• Figure 3: Diff-1 problem: Partition of $\mathrm{Y}^{(N)}$ obtained with algorithm $\mathcal{M}\texttt{-based-split}$ after 2 iterations (left), 4 iterations (middle) and 10 iterations (right) in the case $\alpha=0.105$.
• Figure 4: Diff-1 problem: Final partition of $\mathrm{Y}^{(N)}$ generated by the algorithm $\mathrm{Y}\texttt{-cart-split}$ with $n_{\max}=10$ when $\alpha=0.105$. The local spaces in the library are of dimension $6$ (orange), $7$ (green), $8$ (blue) and $10$ (red).
• Figure 5: Diff-2 problem: Plot of the functions defined \ref{['eqn:diff_a_pw_part1']} and \ref{['eqn:diff_a_pw_part2']}.

Theorems & Definitions (3)

• remark 1
• remark 2
• remark 3