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Leveraging time and parameters for nonlinear model reduction methods

Silke Glas, Benjamin Unger

TL;DR

This paper tackles the challenge of model order reduction for problems with slowly decaying Kolmogorov $n$-widths by using nonlinear projections implemented via autoencoders. The authors propose augmenting the training data with explicit time and parameter coordinates, and extend the dynamical system to include these coordinates so that a linear encoder suffices. They show theoretically and numerically that when the reduced dimension is $r = p+1$, the extended formulation achieves the same approximation power as the original nonlinear approach while reducing hyperparameters. Numerical experiments on Burgers' equation and a transport-dominated advection equation demonstrate that time-extended linear-encoder autoencoders can match or outperform nonlinear variants, with faster training times and robust performance.

Abstract

In this paper, we consider model order reduction (MOR) methods for problems with slowly decaying Kolmogorov $n$-widths as, e.g., certain wave-like or transport-dominated problems. To overcome this Kolmogorov barrier within MOR, nonlinear projections are used, which are often realized numerically using autoencoders. These autoencoders generally consist of a nonlinear encoder and a nonlinear decoder and involve costly training of the hyperparameters to obtain a good approximation quality of the reduced system. To facilitate the training process, we show that extending the to-be-reduced system and its corresponding training data makes it possible to replace the nonlinear encoder with a linear encoder without sacrificing accuracy, thus roughly halving the number of hyperparameters to be trained.

Leveraging time and parameters for nonlinear model reduction methods

TL;DR

This paper tackles the challenge of model order reduction for problems with slowly decaying Kolmogorov -widths by using nonlinear projections implemented via autoencoders. The authors propose augmenting the training data with explicit time and parameter coordinates, and extend the dynamical system to include these coordinates so that a linear encoder suffices. They show theoretically and numerically that when the reduced dimension is , the extended formulation achieves the same approximation power as the original nonlinear approach while reducing hyperparameters. Numerical experiments on Burgers' equation and a transport-dominated advection equation demonstrate that time-extended linear-encoder autoencoders can match or outperform nonlinear variants, with faster training times and robust performance.

Abstract

In this paper, we consider model order reduction (MOR) methods for problems with slowly decaying Kolmogorov -widths as, e.g., certain wave-like or transport-dominated problems. To overcome this Kolmogorov barrier within MOR, nonlinear projections are used, which are often realized numerically using autoencoders. These autoencoders generally consist of a nonlinear encoder and a nonlinear decoder and involve costly training of the hyperparameters to obtain a good approximation quality of the reduced system. To facilitate the training process, we show that extending the to-be-reduced system and its corresponding training data makes it possible to replace the nonlinear encoder with a linear encoder without sacrificing accuracy, thus roughly halving the number of hyperparameters to be trained.
Paper Structure (8 sections, 21 equations, 5 figures, 5 tables)

This paper contains 8 sections, 21 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Setting and known work
  3. Contribution
  4. Numerical examples
  5. Autoencoder Training
  6. Burgers' equation
  7. Advection equation
  8. Conclusion

Figures (5)

  • Figure 1: Burgers' equation: solution at selected times.
  • Figure 2: Burgers' equation: approximations (top) and corresponding errors in absolute value (bottom) for the solution at $t=0.03$ over different autoencoders with $r=1$ and scenario (A). The first 4 lines in the top figure nearly coincide.
  • Figure 3: Burgers' equation: POD (solid) and NLA (dashed) approximation for times $t_1= 1.35$, $t_2=4.00$.
  • Figure 4: Advection equation: approximations (top) and corresponding errors in absolute values (bottom) for the solution at $t=0.49$ over different autoencoders with $r=1$ and scenario (A).
  • Figure 5: Advection equation: point projection property \ref{['eqn:projectionProperty']} for the trained autoencoders NNA, $\text{LNA\xspace}_{\text{ext}}$, LNA, $\text{LNA\xspace}_{\text{ext,fix}}$$\,$with $r=1$ and scenario (A).