RRAEDy: Adaptive Latent Linearization of Nonlinear Dynamical Systems

2025-12-08

AI Summary

Existing latent-space models for dynamical systems often require manual specification of latent dimensionality, rely on intricate loss balancing to approximate linear dynamics, and lack effective regularization of latent variables. To address these limitations, we propose the Rank-Reduced Autoencoder (RRAE), which automatically determines the optimal latent dimension via singular-value ordering and pruning—eliminating manual tuning. By integrating low-rank constraints with Dynamic Mode Decomposition (DMD), RRAE jointly enforces linearization, stability guarantees, and interpretability in the learned latent space. This design circumvents conventional multi-objective loss trade-offs, significantly improving modeling robustness. Evaluated on canonical nonlinear benchmarks—including the Van der Pol oscillator, Burgers’ equation, 2D Navier–Stokes flow, and rotating Gaussian dynamics—RRAE achieves high-fidelity long-term predictions and strong generalization across diverse regimes.

Technology Category

• Machine Learning: Deep Generative Models & Autoencoders
• Intelligent Robots: State Estimation
• Computer Vision: Diffusion Models for Vision

Application Category

• Search and Retrieval-Augmented AI: Web search models and ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Determines optimal latent dimension automatically for dynamical systems
• Enforces linearized and regularized latent dynamics without complex loss balancing
• Learns stable low-dimensional dynamics without auxiliary losses or manual tuning

Innovation

• Automatically discovers and prunes latent dimension via singular values
• Learns latent linear dynamics using Dynamic Mode Decomposition operator
• Enforces stability without auxiliary losses or manual tuning