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RRAEDy: Adaptive Latent Linearization of Nonlinear Dynamical Systems

Jad Mounayer, Sebastian Rodriguez, Jerome Tomezyk, Chady Ghnatios, Francisco Chinesta

TL;DR

RRAEDy introduces an adaptive latent-dimension framework for nonlinear dynamical systems by coupling Rank Reduction Autoencoders with a Dynamic Mode Decomposition operator to enforce regularized, linear latent dynamics. The method automatically prunes latent variables via truncated SVD and uses a fixed basis with a learned DMD for stable evolution, trained with a single reconstruction loss. The authors provide theoretical results on the stability and batch-consistency of the learned operator and demonstrate strong predictive performance on benchmarks including Van der Pol, Burgers', 2D Navier–Stokes, and Rotating Gaussians, with an extension to parametric ODEs. The work is complemented by open-source code and shows robust extrapolation and resilience to local latent-space behavior, highlighting practical impact for data-driven dynamical modeling.

Abstract

Most existing latent-space models for dynamical systems require fixing the latent dimension in advance, they rely on complex loss balancing to approximate linear dynamics, and they don't regularize the latent variables. We introduce RRAEDy, a model that removes these limitations by discovering the appropriate latent dimension, while enforcing both regularized and linearized dynamics in the latent space. Built upon Rank-Reduction Autoencoders (RRAEs), RRAEDy automatically rank and prune latent variables through their singular values while learning a latent Dynamic Mode Decomposition (DMD) operator that governs their temporal progression. This structure-free yet linearly constrained formulation enables the model to learn stable and low-dimensional dynamics without auxiliary losses or manual tuning. We provide theoretical analysis demonstrating the stability of the learned operator and showcase the generality of our model by proposing an extension that handles parametric ODEs. Experiments on canonical benchmarks, including the Van der Pol oscillator, Burgers' equation, 2D Navier-Stokes, and Rotating Gaussians, show that RRAEDy achieves accurate and robust predictions. Our code is open-source and available at https://github.com/JadM133/RRAEDy. We also provide a video summarizing the main results at https://youtu.be/ox70mSSMGrM.

RRAEDy: Adaptive Latent Linearization of Nonlinear Dynamical Systems

TL;DR

RRAEDy introduces an adaptive latent-dimension framework for nonlinear dynamical systems by coupling Rank Reduction Autoencoders with a Dynamic Mode Decomposition operator to enforce regularized, linear latent dynamics. The method automatically prunes latent variables via truncated SVD and uses a fixed basis with a learned DMD for stable evolution, trained with a single reconstruction loss. The authors provide theoretical results on the stability and batch-consistency of the learned operator and demonstrate strong predictive performance on benchmarks including Van der Pol, Burgers', 2D Navier–Stokes, and Rotating Gaussians, with an extension to parametric ODEs. The work is complemented by open-source code and shows robust extrapolation and resilience to local latent-space behavior, highlighting practical impact for data-driven dynamical modeling.

Abstract

Most existing latent-space models for dynamical systems require fixing the latent dimension in advance, they rely on complex loss balancing to approximate linear dynamics, and they don't regularize the latent variables. We introduce RRAEDy, a model that removes these limitations by discovering the appropriate latent dimension, while enforcing both regularized and linearized dynamics in the latent space. Built upon Rank-Reduction Autoencoders (RRAEs), RRAEDy automatically rank and prune latent variables through their singular values while learning a latent Dynamic Mode Decomposition (DMD) operator that governs their temporal progression. This structure-free yet linearly constrained formulation enables the model to learn stable and low-dimensional dynamics without auxiliary losses or manual tuning. We provide theoretical analysis demonstrating the stability of the learned operator and showcase the generality of our model by proposing an extension that handles parametric ODEs. Experiments on canonical benchmarks, including the Van der Pol oscillator, Burgers' equation, 2D Navier-Stokes, and Rotating Gaussians, show that RRAEDy achieves accurate and robust predictions. Our code is open-source and available at https://github.com/JadM133/RRAEDy. We also provide a video summarizing the main results at https://youtu.be/ox70mSSMGrM.

Paper Structure

This paper contains 22 sections, 5 theorems, 58 equations, 16 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Classical system identification.
  3. Neural approaches.
  4. Latent-space modeling.
  5. Koopman-inspired models.
  6. Our approach.
  7. We summarize our contributions as follows:
  8. Background
  9. Methodology
  10. Model definition for a fixed bottleneck.
  11. Advantages (Theoretical and Practical).
  12. Results
  13. Theoretical Validation - Ablation Study
  14. ODEs with parameters
  15. Conclusion and Perspectives
  16. ...and 7 more sections

Key Result

Lemma 1

When both Hypothesis hyp:1 and Hypothesis hyp:2 hold, the learned DMD operators $W_i$ for different batches $i$ converge to similar operators. In other words, we can write,

Figures (16)

  • Figure 1: Architecture of RRAEDy for one sample ($N=1$).
  • Figure 2: Predictions obtained using RRAEDy on the Van der Pol oscillator dataset (test set, only initial condition is given as input). Phase diagram (to the right), and time series (to the left).
  • Figure 3: Predictions obtained using RRAEDy on the Burger's equation dataset for two initial conditions (test set, only initial condition is given as input).
  • Figure 4: Predictions obtained using RRAEDy on the fluid flow behind a cylinder dataset for a flow with low Reynold's number for different times (test set, only initial condition is given as input).
  • Figure 5: Predictions obtained using RRAEDy on the fluid flow behind a cylinder dataset for a flow with high Reynold's number for different times (test set, only initial condition is given as input).
  • ...and 11 more figures

Theorems & Definitions (5)

  • Lemma 1
  • Lemma 2
  • Lemma
  • Lemma
  • Lemma