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Data-driven discovery of interpretable Lagrangian of stochastically excited dynamical systems

Tapas Tripura, Satyam Panda, Budhaditya Hazra, Souvik Chakraborty

TL;DR

An automated data-driven framework is proposed for the simultaneous yet uncoupled discovery of Lagrange densities and the volatility function of stochastic excitation by leveraging the sparse regression, providing an interpretable description of the underlying Lagrange density, allowing for a deeper understanding of system dynamics under stochastic excitations.

Abstract

Exploring the intersection of deterministic and stochastic dynamics, this paper delves into Lagrangian discovery for conservative and non-conservative systems under stochastic excitation. Traditional Lagrangian frameworks, adept at capturing deterministic behavior, are extended to incorporate stochastic excitation. The study critically evaluates recent computational methodologies for learning Lagrangians from observed data, highlighting the limitations in interpretability and the exclusion of stochastic excitation. To address these gaps, an automated data-driven framework is proposed for the simultaneous yet uncoupled discovery of Lagrange densities and the volatility function of stochastic excitation by leveraging the sparse regression. This novel framework offers several advantages over existing approaches. Firstly, it provides an interpretable description of the underlying Lagrange density, allowing for a deeper understanding of system dynamics under stochastic excitations. Secondly, it identifies the interpretable form of the generalized stochastic force, addressing the limitations of existing deterministic approaches. Additionally, the framework demonstrates robustness and versatility through numerical case studies encompassing both stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs), with results showing almost exact approximations to true system behavior and minimal relative error in derived equations of motion.

Data-driven discovery of interpretable Lagrangian of stochastically excited dynamical systems

TL;DR

An automated data-driven framework is proposed for the simultaneous yet uncoupled discovery of Lagrange densities and the volatility function of stochastic excitation by leveraging the sparse regression, providing an interpretable description of the underlying Lagrange density, allowing for a deeper understanding of system dynamics under stochastic excitations.

Abstract

Exploring the intersection of deterministic and stochastic dynamics, this paper delves into Lagrangian discovery for conservative and non-conservative systems under stochastic excitation. Traditional Lagrangian frameworks, adept at capturing deterministic behavior, are extended to incorporate stochastic excitation. The study critically evaluates recent computational methodologies for learning Lagrangians from observed data, highlighting the limitations in interpretability and the exclusion of stochastic excitation. To address these gaps, an automated data-driven framework is proposed for the simultaneous yet uncoupled discovery of Lagrange densities and the volatility function of stochastic excitation by leveraging the sparse regression. This novel framework offers several advantages over existing approaches. Firstly, it provides an interpretable description of the underlying Lagrange density, allowing for a deeper understanding of system dynamics under stochastic excitations. Secondly, it identifies the interpretable form of the generalized stochastic force, addressing the limitations of existing deterministic approaches. Additionally, the framework demonstrates robustness and versatility through numerical case studies encompassing both stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs), with results showing almost exact approximations to true system behavior and minimal relative error in derived equations of motion.
Paper Structure (16 sections, 34 equations, 4 figures, 3 tables, 3 algorithms)

This paper contains 16 sections, 34 equations, 4 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Lagrangian discovery
  4. Discovery of Lagrangian functional
  5. Discovery of volatility of the stochastic excitation
  6. Numerical experiments
  7. Stochastic Harmonic oscillator
  8. Stochastic pendulum
  9. Stochastic Duffing oscillator
  10. Stochastic 3DOF oscillator
  11. Stochastic Wave equation
  12. Stochastic Euler-Bernoulli Beam
  13. Identification of SDEs from the discovered Lagrangian
  14. Identifying Hamiltonian representations from the discovered Lagrangian
  15. Conclusions
  16. ...and 1 more sections

Figures (4)

  • Figure 1: Discovery of the Lagrangian basis functions for the example SDEs and SPDEs. The identified basis functions are shown in the x-axis, and the associated parameters are shown in the y-axis. (a) The plots on the left side show the identified basis functions of deterministic Lagrangian density. The library matrix $\mathbf{D}^{\text{f}}$ for the deterministic Lagrangian density contains 25 basis functions for the stochastic Harmonic oscillator and stochastic pendulum, 15 basis functions for the stochastic Duffing oscillator, 50 for the stochastic 3DOF structural system, 254 for the stochastic wave equation, and 421 for the stochastic Euler-Bernoulli Beam. (b) The plots on the right show the potential energy basis functions associated with the diffusion process. In this case, the library matrix $\mathbf{D}^{\text{g}}$ contains 12 basis functions for the stochastic harmonic, pendulum, and Duffing oscillator, 17 for the 3DOF system, 204 for the stochastic wave equation, and 200 for the stochastic Euler-Bernoulli Beam equation.
  • Figure 2: Time evolution of the responses of the true and identified stochastic differential equations. The solid line indicates the expected response of the true (red solid line) and discovered (dashed blue line) SDEs. The green shaded region indicates the uncertainty (two times the standard deviation) in the system response due to the stochastic Wiener process. The black vertical solid line separates the training (the left side) and prediction regions (the right side).
  • Figure 3: Time evolution of the response of the discovered wave equation and Euler-Bernoulli beam. The ground truth indicates the response of the true stochastic wave equation and stochastic Euler-Bernoulli Beam. The prediction indicates the response of the discovered stochastic wave equation and stochastic Euler-Bernoulli Beam. The predictive error indicates the absolute error between the ground truth and the predicted response. The standard deviation indicates the randomness in the system response due to the stochastic Wiener noise. The left side of the white vertical indicates the training duration, and the right side indicates the prediction duration.
  • Figure 4: Time evolution of Hamiltonian of the example problems. The solid lines indicate the Hamiltonian trajectory of the true systems, and the dashlines indicate the Hamiltonian trajectory of the discovered systems.