自动发现最优隐空间维度，让动态系统建模更稳定准确
RRAEDy: Adaptive Latent Linearization of Nonlinear Dynamical Systems
- 通过奇异值排序自动剪枝隐变量，自适应确定隐空间维数
- 无需额外损失函数，在隐空间实现线性化与正则化动态建模
- 适合需要稳定低维建模的物理系统仿真与预测任务
现有隐空间动态系统模型通常需预先设定隐维数，依赖复杂损失平衡来近似线性动力学，且缺乏对隐变量的正则化。本文提出RRAEDy，基于秩缩减自编码器（RRAE），通过奇异值自动排序并剪枝隐变量，同时学习一个控制其时间演化的隐空间动态模态分解（DMD）算子。该结构无约束但具线性约束，使模型在无需辅助损失或人工调参的情况下，学习到稳定且低维的动力学。理论分析证明了所学算子的稳定性，并扩展至处理参数化常微分方程。在范德波尔振子、伯格斯方程、二维纳维-斯托克斯方程及旋转高斯等经典基准上实验表明，RRAEDy实现了准确且鲁棒的预测。代码已开源，详见https://github.com/JadM133/RRAEDy，视频演示见https://youtu.be/ox70mSSMGrM。
原文摘要 · Abstract (English)
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.
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