Learning Continuous and Discrete Variational ODEs with Convergence Guarantee and Uncertainty Quantification

Conceitos Básicos

A method is introduced to learn continuous or discrete Lagrangian dynamics from data, with guaranteed convergence as the distance between observation points goes to zero, and the ability to quantify uncertainty in the learned models.

Resumo

The article presents a framework for learning continuous and discrete Lagrangian dynamics from data using Gaussian processes. The key highlights are:

The method learns a Lagrangian function L or discrete Lagrangian Ld that governs the dynamics of the system, such that the Euler-Lagrange equations EL(L) = 0 or discrete Euler-Lagrange equations DEL(Ld) = 0 are satisfied at the observed data points.

The learned Lagrangian or discrete Lagrangian is the conditional mean of a Gaussian process, which provides a guarantee of convergence to the true Lagrangian as the distance between observation points goes to zero.

The framework allows for efficient uncertainty quantification of any linear observable of the Lagrangian system, such as the Hamiltonian function (energy) and symplectic structure. This can enable adaptive sampling techniques to improve the model.

The method addresses the inherent ambiguity in identifying Lagrangians from data through careful regularization strategies, exploiting the connection between Gaussian process regression and constrained optimization problems in reproducing kernel Hilbert spaces.

Numerical experiments on a coupled harmonic oscillator demonstrate the convergence properties of the method and the ability to quantify uncertainty in the learned models.

Estatísticas

The article presents the following key figures and metrics:

Variance of the Euler-Lagrange residual EL(ξM) for M=80 and M=300 data points, showing decreasing uncertainty as more data is used.
Comparison of a computed motion using the learned Lagrangian L300 versus the true reference solution, demonstrating qualitative agreement.
Plots of the learned Hamiltonian HM and its standard deviation σHM, showing decreasing uncertainty as M increases.
Error in predicting the acceleration Accx(LM) versus the true Accx(Lref), demonstrating convergence as M increases.
Convergence plot of the relative error in predicted acceleration errAcc(x) for the 1D harmonic oscillator, showing convergence to machine precision.

Citações

"The article introduces a method to learn dynamical systems that are governed by Euler–Lagrange equations from data."
"The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore, structure preserving by design."
"The article overcomes major practical and theoretical difficulties related to the ill-posedness of the identification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex minimisation problems in reproducing kernel Hilbert spaces."

Principais Insights Extraídos De

Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification

by Christian Of... às arxiv.org 05-01-2024

https://arxiv.org/pdf/2404.19626.pdf

Perguntas Mais Profundas

How can the proposed framework be extended to learn Lagrangian dynamics in the presence of external forces or constraints

To extend the proposed framework to learn Lagrangian dynamics in the presence of external forces or constraints, we can incorporate these additional factors into the learning process. External forces can be included as additional terms in the Lagrangian function, allowing the model to account for their effects on the system dynamics. Constraints can be integrated into the learning algorithm by imposing conditions that the Lagrangian must satisfy to ensure that the learned model adheres to the constraints. By including these elements in the training data and adjusting the optimization process accordingly, the framework can be adapted to handle more complex dynamical systems with external influences and constraints.

What are the implications of the ambiguity of Lagrangians on the interpretability and generalization of the learned models

The ambiguity of Lagrangians poses challenges for the interpretability and generalization of the learned models. Since multiple Lagrangians can lead to the same equations of motion, it becomes difficult to uniquely determine the underlying dynamics of the system based on the observed data alone. This ambiguity can impact the reliability of the learned models, as different Lagrangians may yield different predictions for future behavior. To address this issue, regularization strategies and normalization techniques can be employed to improve the conditioning of the identified Lagrangians and enhance the interpretability and generalization capabilities of the models. By carefully managing the ambiguity of Lagrangians, we can mitigate the risks of model inaccuracies and improve the robustness of the learned dynamical systems.

Can the techniques developed in this work be applied to learn other types of physical models, such as Hamiltonian or port-Hamiltonian systems, from data

The techniques developed in this work can be applied to learn other types of physical models, such as Hamiltonian or port-Hamiltonian systems, from data. By adapting the framework to accommodate the specific structures and constraints of these systems, we can leverage Gaussian process regression and uncertainty quantification methods to identify the underlying dynamics. For Hamiltonian systems, the focus would be on learning the Hamiltonian function that governs the system's evolution, while for port-Hamiltonian systems, the emphasis would be on capturing the energy-based modeling approach and the associated structure-preserving properties. By tailoring the learning process to the characteristics of these systems and incorporating relevant observables and constraints, we can extend the applicability of the developed techniques to a broader range of physical models.