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A Robust SINDy Autoencoder for Noisy Dynamical System Identification

Kairui Ding

Abstract

Sparse identification of nonlinear dynamics (SINDy) has been widely used to discover the governing equations of a dynamical system from data. It uses sparse regression techniques to identify parsimonious models of unknown systems from a library of candidate functions. Therefore, it relies on the assumption that the dynamics are sparsely represented in the coordinate system used. To address this limitation, one seeks a coordinate transformation that provides reduced coordinates capable of reconstructing the original system. Recently, SINDy autoencoders have extended this idea by combining sparse model discovery with autoencoder architectures to learn simplified latent coordinates together with parsimonious governing equations. A central challenge in this framework is robustness to measurement error. Inspired by noise-separating neural network structures, we incorporate a noise-separation module into the SINDy autoencoder architecture, thereby improving robustness and enabling more reliable identification of noisy dynamical systems. Numerical experiments on the Lorenz system show that the proposed method recovers interpretable latent dynamics and accurately estimates the measurement noise from noisy observations.

A Robust SINDy Autoencoder for Noisy Dynamical System Identification

Abstract

Sparse identification of nonlinear dynamics (SINDy) has been widely used to discover the governing equations of a dynamical system from data. It uses sparse regression techniques to identify parsimonious models of unknown systems from a library of candidate functions. Therefore, it relies on the assumption that the dynamics are sparsely represented in the coordinate system used. To address this limitation, one seeks a coordinate transformation that provides reduced coordinates capable of reconstructing the original system. Recently, SINDy autoencoders have extended this idea by combining sparse model discovery with autoencoder architectures to learn simplified latent coordinates together with parsimonious governing equations. A central challenge in this framework is robustness to measurement error. Inspired by noise-separating neural network structures, we incorporate a noise-separation module into the SINDy autoencoder architecture, thereby improving robustness and enabling more reliable identification of noisy dynamical systems. Numerical experiments on the Lorenz system show that the proposed method recovers interpretable latent dynamics and accurately estimates the measurement noise from noisy observations.

Paper Structure

This paper contains 47 sections, 1 theorem, 48 equations, 5 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Center manifold theory and normal forms
  4. Center manifold theorem
  5. Normal form theory
  6. Combining center manifold and normal form theories
  7. SINDy
  8. Multi-layer perceptrons and autoencoders
  9. Neural networks and multi-layer perceptrons
  10. Autoencoders
  11. Noise-separating network
  12. Methods
  13. Problem formulation
  14. Data cleansing
  15. Autoencoder and optimization procedure
  16. ...and 32 more sections

Key Result

Theorem 2.1

Let $A$ be the Jacobian matrix of the system at the equilibrium point $0$, and let $E^s$, $E^u$, and $E^c$ be the stable, unstable, and center subspaces of $\mathbb{R}^n$, respectively. Then: $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: Schematic of a simple Multi-Layer Perceptron (MLP)
  • Figure 2: Schematic of an Autoencoder
  • Figure 3: Schematic of the robust SINDy autoencoder.
  • Figure 4: Comparison between the learned latent dynamics and the true Lorenz system.
  • Figure 5: Comparison between the learned measurement error and the true injected noise.

Theorems & Definitions (2)

  • Definition 2.1: Center manifold
  • Theorem 2.1: Center Manifold Theorem