Nonlinear manifold approximation using compositional polynomial networks

TL;DR

The paper tackles nonlinear manifold approximation for sets $M$ in a Hilbert space by coupling a linear encoder with a nonlinear, compositional polynomial decoder to yield a low-dimensional manifold $M_n=D(\mathbb{R}^n)$. It develops a rigorous error and stability framework, including mean-squared and worst-case settings, and proposes an adaptive algorithm to construct the encoder/decoder pair with prescribed accuracy and Lipschitz constants. The decoder uses a tree-structured composition of polynomials, with coefficient maps $g_i$ that depend on previously computed coefficients, enabling efficient learning from samples. Numerical experiments on KdV, Allen-Cahn, and Burgers equations demonstrate substantial accuracy gains over state-of-the-art nonlinear MOR methods, illustrating the method’s potential for offline-online model reduction and inverse problems. The work advances nonlinear manifold learning by providing a practical, stable, and data-driven approach that leverages linear encoders and structured polynomial decoders with adaptive subspace selection.

Abstract

We consider the problem of approximating a subset $M$ of a Hilbert space $X$ by a low-dimensional manifold $M_n$, using samples from $M$. We propose a nonlinear approximation method where $M_n $ is defined as the range of a smooth nonlinear decoder $D$ defined on $\mathbb{R}^n$ with values in a possibly high-dimensional linear space $X_N$, and a linear encoder $E$ which associates to an element from $ M$ its coefficients $E(u)$ on a basis of a $n$-dimensional subspace $X_n \subset X_N$, where $X_N$ is an optimal or near to optimal linear space, depending on the selected error measure The linearity of the encoder allows to easily obtain the parameters $E(u)$ associated with a given element $u$ in $M$. The proposed decoder is a polynomial map from $\mathbb{R}^n$ to $X_N$ which is obtained by a tree-structured composition of polynomial maps, estimated sequentially from samples in $M$. Rigorous error and stability analyses are provided, as well as an adaptive strategy for constructing the subspace $X_n$, and a decoder that guarantees an approximation of the set $M$ with controlled mean-squared or wort-case errors, and a controlled stability (Lipschitz continuity) of the encoder and decoder pair. We demonstrate the performance of our method through numerical experiments.

Figures (9)

• Figure 1: Illustrative example for a one-dimensional manifold in $\mathbb{R}^3$.
• Figure 2: (KdV) Predictions for different methods, with $n = 2$.
• Figure 3: (KdV) Comparison between methods for two snapshots, with $n = 2$.
• Figure 4: (KdV) Coefficient-wise errors for Sparse and CPN-S, with $p=5$ and $\epsilon=10^{-4}$.
• Figure 5: (KdV) Compositional networks for different coefficients, using CPN-S with $\epsilon=10^{-4}$ and $p=5$.

Theorems & Definitions (10)

• Remark 2.1
• Remark 2.2
• Lemma 3.1
• Proposition 3.2
• Proposition 3.3
• proof
• Remark 3.4: Estimation of Lipschitz norms
• Remark 4.1
• Remark 4.2
• Remark 4.3