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Updated: Aug 15, 2025

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Learning chaotic systems from noisy data via multi-step optimization and adaptive training.

Lei Zhang1, Shaoqiang Tang2, Guowei He1

  • 1The State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China.

Chaos (Woodbury, N.Y.)
|January 1, 2023
PubMed
Summary
This summary is machine-generated.

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A new Multi-Step-Accumulation (MSA) method discovers governing equations from noisy data by minimizing errors across multiple time steps. This data-driven approach enhances accuracy and robustness compared to one-step models.

Area of Science:

  • Scientific Computing
  • Dynamical Systems Theory
  • Data-Driven Science

Background:

  • Discovering governing equations from experimental data is crucial for scientific understanding.
  • Conventional sparse regression methods like SINDy are limited by their one-step prediction nature and sensitivity to noise.
  • Accurate equation discovery from noisy, incomplete, or irregularly sampled data remains a significant challenge.

Purpose of the Study:

  • To develop a novel data-driven sparse identification method for discovering underlying governing equations from noisy measurement data.
  • To introduce the Multi-Step-Accumulation (MSA) error minimization approach, enhancing robustness against noise and data imperfections.
  • To demonstrate the effectiveness and accuracy of the MSA method across various complex dynamical systems.

Main Methods:

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  • Utilizing sparse representation and a library of candidate functions to identify parsimonious governing equations.
  • Implementing a multi-step prediction model within the Multi-Step-Accumulation (MSA) framework to accumulate errors over time.
  • Employing a loss function that minimizes the total error across all time steps between measured and predicted data series.
  • Developing an adaptive training strategy to gradually increase time series length for improved nonlinear optimization and prediction accuracy.

Main Results:

  • The proposed MSA method demonstrates superior robustness and accuracy in identifying governing equations from noisy data compared to conventional one-step methods.
  • Successful application of MSA to diverse systems, including chaotic maps, damped oscillators, the Lorenz system, and turbulent flow models.
  • Validation of MSA's performance under challenging conditions: noisy measurements, missing data, and large time step sizes.
  • The adaptive training strategy significantly improves prediction accuracy, particularly for chaotic systems.

Conclusions:

  • The Multi-Step-Accumulation (MSA) method offers a powerful and robust approach for data-driven discovery of governing equations from noisy measurements.
  • MSA's multi-step prediction strategy effectively mitigates noise corruption and improves the identification of complex dynamical systems.
  • The adaptive training strategy further enhances the method's applicability and accuracy, addressing nonlinear optimization challenges.