Data-driven Identification of Parametric Governing Equations of Dynamical Systems Using the Signed Cumulative...

Area of Science:

• Dynamical Systems Analysis
• Applied Mathematics
• Computational Engineering

Background:

• Accurate identification of partial differential equation (PDE) parameters is crucial for understanding and modeling dynamical systems.
• Estimating parameters, especially those related to structural damage, often involves complex nonlinear regression problems.
• Existing methods may require prior knowledge of system excitation or initial conditions, limiting their practical application.

Purpose of the Study:

• To present a novel data-driven approach for identifying PDE parameters in dynamical systems.
• To develop a method capable of estimating parameters, including those indicative of structural damage, without requiring a priori knowledge of excitation or initial conditions.
• To demonstrate the effectiveness of the proposed technique in system identification and structural health monitoring (SHM).

Main Methods:

• A mathematical "transport" model is employed for the dynamical system's solution at specific spatial locations.
• A newly developed mathematical transform, the signed cumulative distribution transform (SCDT), is utilized.
• The SCDT converts the nonlinear parameter estimation problem into a simple linear regression, enabling parameter recovery from solutions measured at a single location using training data.

Main Results:

• The proposed signed cumulative distribution transform (SCDT) effectively transforms nonlinear parameter estimation into linear regression.
• Numerical experiments demonstrate superior accuracy in detecting and estimating PDE parameters compared to recent machine learning methods.
• A damage identification experiment on a public dataset confirms the method's high effectiveness for structural health monitoring (SHM) applications.

Conclusions:

• The developed data-driven approach provides an accurate and efficient method for PDE parameter identification.
• The technique's ability to work without prior knowledge of excitation or initial conditions enhances its practical utility.
• The method shows significant promise for applications in structural health monitoring and other fields requiring dynamical system analysis.