[Paper Review] Practicable Simulation-Free Model Order Reduction by Nonlinear Moment Matching

María Cruz de Carlos Varona, Raphael Gebhart+2 more|arXiv (Cornell University)|Jan 30, 2019

Model Reduction and Neural Networks29 references4 citations

TL;DR

This paper presents a simulation-free, numerically efficient model order reduction method for nonlinear systems using nonlinear moment matching. By replacing the computationally expensive solution of a nonlinear partial differential equation with the solution of nonlinear algebraic equations (NLSEs), the method enables feasible, high-fidelity reduced-order models without requiring full-order simulations during training.

ABSTRACT

In this paper, a practicable simulation-free model order reduction method by nonlinear moment matching is developed. Based on the steady-state interpretation of linear moment matching, we comprehensively explain the extension of this reduction concept to nonlinear systems presented in [1], provide some new insights and propose some simplifications to achieve a feasible and numerically efficient nonlinear model reduction algorithm. This algorithm relies on the solution of nonlinear systems of equations rather than on the expensive simulation of the original model or the difficult solution of a nonlinear partial differential equation.

Motivation & Objective

• To develop a practical, simulation-free model order reduction technique for nonlinear systems.
• To overcome the computational bottleneck of solving nonlinear Sylvester-like PDEs in existing nonlinear moment matching approaches.
• To enable efficient reduced-order modeling by replacing full-order simulations with the solution of nonlinear systems of equations.
• To provide a numerically feasible algorithm that matches nonlinear moments without relying on trajectory-based data.
• To demonstrate the effectiveness of the method on a benchmark nonlinear PDE system (FitzHugh-Nagumo).

Proposed method

• Proposes a nonlinear moment matching framework based on the steady-state response of nonlinear systems to exponential inputs.
• Replaces the solution of a nonlinear partial differential equation with solving a set of nonlinear algebraic equations (NLSEs) derived from a signal generator.
• Uses a linear projection onto a subspace spanned by solutions of NLSEs to construct the reduced-order model.
• Employs a user-defined signal generator to generate input trajectories that capture the system's nonlinear behavior.
• Applies singular value decomposition (SVD) to a smaller matrix of NLSE solutions to extract the reduced basis.
• Extends the method to implicit descriptor systems by working directly with the implicit form rather than inverting the descriptor matrix.

Experimental results

Research questions

• RQ1 Can nonlinear moment matching be made computationally feasible without relying on full-order simulations or solving PDEs?
• RQ2 How can the steady-state response concept from linear systems be generalized to nonlinear systems for model reduction?
• RQ3 What role does the choice of signal generator play in the accuracy and efficiency of the reduced-order model?
• RQ4 Can a simulation-free approach achieve comparable or better accuracy than simulation-based methods like POD?
• RQ5 How does the proposed method compare to data-driven moment estimation techniques in terms of computational cost and fidelity?

Key findings

• The proposed method reduces the training time for a 2000-degree-of-freedom FHN model to 46.17 seconds, compared to 382.25 seconds for POD.
• For a reduced order of 22, the relative L1 error norm of NLMM is 3.36×10⁻³, while POD achieves 1.03×10⁻⁵, indicating a trade-off between speed and accuracy.
• For a higher reduced order of 34, NLMM achieves a relative L1 error of 1.83×10⁻³, showing improved accuracy with more basis vectors.
• The method avoids full-order simulations during training, relying only on solving 41 nonlinear systems of equations.
• The signal generator significantly influences the quality of the reduced model, with unstable exponential inputs effectively capturing system dynamics.
• The approach is applicable to implicit descriptor systems by directly using the implicit form, avoiding numerical issues from matrix inversion.