# Deep polytopic autoencoders for low-dimensional linear parameter-varying approximations and nonlinear feedback design

arXiv: https://arxiv.org/abs/2403.18044 · HTML: https://mlsanity.com/paper/2403.18044/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 50 references that are in the map.
- First posted on arXiv 26 Mar 2024.

## Builds on (most cited first, 16 of 16)

- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021)
- [Deep k-Means: Jointly clustering with k-Means and learning representations](https://mlsanity.com/paper/1806.10069/) (2018)
- [High-order Differentiable Autoencoder for Nonlinear Model Reduction](https://mlsanity.com/paper/2102.11026/) (2021)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures](https://mlsanity.com/paper/1910.09045/) (2019)
- [State-dependent Riccati equation feedback stabilization for nonlinear PDEs](https://mlsanity.com/paper/2106.07163/) (2021)
- [Convolutional Autoencoders, Clustering and POD for Low-dimensional Parametrization of Navier-Stokes Equations](https://mlsanity.com/paper/2302.01278/) (2023)
- [Polytopic Autoencoders with Smooth Clustering for Reduced-order Modelling of Flows](https://mlsanity.com/paper/2401.10620/) (2024)
- [A POD approach to identify and control PDEs online through State Dependent Riccati equations](https://mlsanity.com/paper/2402.08186/) (2024)
- [Exponential Stability and Stabilization of Extended Linearizations via Continuous Updates of Riccati Based Feedback](https://mlsanity.com/paper/1607.08441/) (2016)
- [Singular value decay of operator-valued differential Lyapunov and Riccati equations](https://mlsanity.com/paper/1804.02197/) (2018)
- [Feedback Stabilization of the Two-Dimensional Navier-Stokes Equations by Value Function Approximation](https://mlsanity.com/paper/1902.00394/) (2019)
- [A HJB-POD approach for the control of nonlinear PDEs on a tree structure](https://mlsanity.com/paper/1905.03395/) (2019)
- [Low-complexity linear parameter-varying approximations of incompressible Navier-Stokes equations for truncated state-dependent Riccati feedback](https://mlsanity.com/paper/2303.11515/) (2023)
- [Using LDLT factorizations in Newton's method for solving general large-scale algebraic Riccati equations](https://mlsanity.com/paper/2402.06844/) (2024)

## Abstract (arXiv)

Polytopic autoencoders provide low-di-men-sion-al parametrizations of states in a polytope. For nonlinear PDEs, this is readily applied to low-dimensional linear parameter-varying (LPV) approximations as they have been exploited for efficient nonlinear controller design via series expansions of the solution to the state-dependent Riccati equation. In this work, we develop a polytopic autoencoder for control applications and show how it improves on standard linear approaches in view of LPV approximations of nonlinear systems. We discuss how the particular architecture enables exact representation of target states and higher order series expansions of the nonlinear feedback law at little extra computational effort in the online phase and how the linear though high-dimensional and nonstandard Lyapunov equations are efficiently computed during the offline phase. In a numerical study, we illustrate the procedure and how this approach can reliably outperform the standard linear-quadratic regulator design.