[Paper Review] A registration method for model order reduction: data compression and geometry reduction

Tommaso Taddei|arXiv (Cornell University)|Jun 26, 2019

Model Reduction and Neural NetworksPhysics and Astronomy62 references87 citations

TL;DR

The paper introduces a general, equation-agnostic registration method for parameterized model order reduction that builds a parameter-dependent bijection to map solutions (or geometries) into a form more amenable to linear compression, with theoretical guarantees and 2D numerical demonstrations.

ABSTRACT

We propose a general --- i.e., independent of the underlying equation --- registration method for parameterized Model Order Reduction. Given the spatial domain $\\Omega \\subset \\mathbb{R}^d$ and a set of snapshots $\\{ u^k \\}_{k=1}^{n_{\ m train}}$ over $\\Omega$ associated with $n_{\ m train}$ values of the model parameters $\\mu^1,\\ldots, \\mu^{n_{\ m train}} \\in \\mathcal{P}$, the algorithm returns a parameter-dependent bijective mapping $\\boldsymbol{\\Phi}: \\Omega \ imes \\mathcal{P} \ o \\mathbb{R}^d$: the mapping is designed to make the mapped manifold $\\{ u_{\\mu} \\circ \\boldsymbol{\\Phi}_{\\mu}: \\, \\mu \\in \\mathcal{P} \\}$ more suited for linear compression methods. We apply the registration procedure, in combination with a linear compression method, to devise low-dimensional representations of solution manifolds with slowly-decaying Kolmogorov $N$-widths; we also consider the application to problems in parameterized geometries. We present a theoretical result to show the mathematical rigor of the registration procedure. We further present numerical results for several two-dimensional problems, to empirically demonstrate the effectivity of our proposal.

Motivation & Objective

• Develop a registration framework that yields a parameter-dependent bijection Φ that maps the solution/geometry to a space better suited for linear compression.
• Establish theoretical guarantees for the registration mapping and its invertibility under specified conditions.
• Demonstrate two applications—data compression of solution fields and geometry reduction for parameterized domains—through numerical experiments in 2D.
• Integrate the registration approach with standard pMOR offline/online paradigms and POD-based compression.

Proposed method

• Define a parametric mapping Φ: Ω × P → R^d that is a bijection for all μ ∈ P.
• Formulate a nonlinear, nonconvex optimization to minimize a reference-mapped-field discrepancy using a reference μ̄ and a set of snapshots.
• Represent Φ as Ψ_a^hf(X) = X + ∑_{m=1}^{M_hf} a_m φ_m^hf(X) with φ_m^hf ∈ Lip(U, R^d) and enforce invertibility heuristically via a constraint on the Jacobian det(J).
• Use kernel regression to generate a mapping Φ_μ for all μ ∈ P based on offline training data.
• Apply the registration to both the data (mapped solution) and the geometry (mapped domain) to enable linear reductions (e.g., POD) on the mapped manifold.
• Embed the approach in the offline/online pMOR paradigm with a POD-based reduced space and a regression-based online predictor.

(a)

Experimental results

Research questions

• RQ1 Can a parameter-dependent registration map Φ be constructed to minimize the discrepancy between mapped snapshots and a reference field?
• RQ2 Under what conditions is the registration map bijective and invertible, ensuring a well-posed mapped problem?
• RQ3 How can the mapping improve the linear reducibility of parameterized solution manifolds with slowly decaying Kolmogorov N-widths?
• RQ4 Is the registration framework effective for both data compression of solutions and geometry reduction in parameterized domains?

Key findings

• The registration procedure yields a parameter-dependent bijection that maps Ω onto itself (or Ω onto Ω_μ) to improve linear compressibility.
• Theoretical results establish conditions ensuring bijectivity and invertibility of the mapping, with a concrete constructive form for the mapping.
• Numerical experiments in two dimensions demonstrate the method's effectiveness for data compression and geometry reduction.
• The approach integrates with POD-based compression and the offline/online pMOR workflow, offering a general, equation-independent registration step.
• The paper discusses connections to optimal transport and other nonlinear compression ideas, grounding the method in a solid theoretical framework.

(b)