# Leveraging time and parameters for nonlinear model reduction methods

arXiv: https://arxiv.org/abs/2501.03853 · HTML: https://mlsanity.com/paper/2501.03853/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 24 references that are in the map.
- First posted on arXiv 7 Jan 2025.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder](https://mlsanity.com/paper/2009.11990/) (2020)
- [Compute and Energy Consumption Trends in Deep Learning Inference](https://mlsanity.com/paper/2109.05472/) (2021)
- [Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces](https://mlsanity.com/paper/2104.02746/) (2021)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [Kolmogorov n-widths for linear dynamical systems](https://mlsanity.com/paper/1806.09880/) (2018)
- [Decomposition of flow data via gradient-based transport optimization](https://mlsanity.com/paper/2107.03481/) (2021)
- [Optimal control for a class of linear transport-dominated systems via the shifted proper orthogonal decomposition](https://mlsanity.com/paper/2412.18950/) (2024)

## Built on by (most cited first, 1 of 1)

- [Nonlinear model reduction for transport-dominated problems](https://mlsanity.com/paper/2602.01397/) (2026)

## Abstract (arXiv)

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.