Structure-Preserving Learning Using Gaussian Processes and Variational Integrators

• URL: http://arxiv.org/abs/2112.05451v1
• Date: Fri, 10 Dec 2021 11:09:29 GMT
• Title: Structure-Preserving Learning Using Gaussian Processes and Variational Integrators
• Authors: Jan Br\"udigam, Martin Schuck, Alexandre Capone, Stefan Sosnowski, Sandra Hirche
• Abstract summary: We propose the combination of a variational integrator for the nominal dynamics of a mechanical system and learning residual dynamics with Gaussian process regression. We extend our approach to systems with known kinematic constraints and provide formal bounds on the prediction uncertainty.
• Score: 62.31425348954686
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Gaussian process regression is often applied for learning unknown systems and specifying the uncertainty of the learned model. When using Gaussian process regression to learn unknown systems, a commonly considered approach consists of learning the residual dynamics after applying some standard discretization, which might however not be appropriate for the system at hand. Variational integrators are a less common yet promising approach to discretization, as they retain physical properties of the underlying system, such as energy conservation or satisfaction of explicit constraints. In this work, we propose the combination of a variational integrator for the nominal dynamics of a mechanical system and learning residual dynamics with Gaussian process regression. We extend our approach to systems with known kinematic constraints and provide formal bounds on the prediction uncertainty. The simulative evaluation of the proposed method shows desirable energy conservation properties in accordance with the theoretical results and demonstrates the capability of treating constrained dynamical systems.

Related papers

• Learning Controlled Stochastic Differential Equations [61.82896036131116]
  This work proposes a novel method for estimating both drift and diffusion coefficients of continuous, multidimensional, nonlinear controlled differential equations with non-uniform diffusion. We provide strong theoretical guarantees, including finite-sample bounds for (L2), (Linfty), and risk metrics, with learning rates adaptive to coefficients' regularity. Our method is available as an open-source Python library.
  arXiv  Detail & Related papers  (2024-11-04T11:09:58Z)
• Dynamic Bayesian Learning and Calibration of Spatiotemporal Mechanistic System [0.0]
  We develop an approach for fully learning and calibration of mechanistic models based on noisy observations. We demonstrate this flexibility through solving problems arising in the analysis of ordinary and partial nonlinear differential equations.
  arXiv  Detail & Related papers  (2022-08-12T23:17:46Z)
• Capturing Actionable Dynamics with Structured Latent Ordinary Differential Equations [68.62843292346813]
  We propose a structured latent ODE model that captures system input variations within its latent representation. Building on a static variable specification, our model learns factors of variation for each input to the system, thus separating the effects of the system inputs in the latent space.
  arXiv  Detail & Related papers  (2022-02-25T20:00:56Z)
• Decimation technique for open quantum systems: a case study with driven-dissipative bosonic chains [62.997667081978825]
  Unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (non-unitary) dynamics. We introduce a method to deal with these systems based on the calculation of (dissipative) lattice Green's function. We illustrate the power of this method with several examples of driven-dissipative bosonic chains of increasing complexity.
  arXiv  Detail & Related papers  (2022-02-15T19:00:09Z)
• Machine learning structure preserving brackets for forecasting irreversible processes [0.0]
  We present a novel parameterization of dissipative brackets from metriplectic dynamical systems. The process learns generalized Casimirs for energy and entropy guaranteed to be conserved and nondecreasing. We provide benchmarks for dissipative systems demonstrating learned dynamics are more robust and generalize better than either "black-box" or penalty-based approaches.
  arXiv  Detail & Related papers  (2021-06-23T18:27:59Z)
• Using Data Assimilation to Train a Hybrid Forecast System that Combines Machine-Learning and Knowledge-Based Components [52.77024349608834]
  We consider the problem of data-assisted forecasting of chaotic dynamical systems when the available data is noisy partial measurements. We show that by using partial measurements of the state of the dynamical system, we can train a machine learning model to improve predictions made by an imperfect knowledge-based model.
  arXiv  Detail & Related papers  (2021-02-15T19:56:48Z)
• Symplectic Gaussian Process Dynamics [5.171909600633905]
  We introduce a sparse process based variational inference scheme that is able to discretize the underlying system with any explicit implicit single or multistep integrator. In particular we discuss Hamiltonian problems coupled with symplectic producing volume preserving predictions.
  arXiv  Detail & Related papers  (2021-02-02T17:02:55Z)
• Learning Constrained Dynamics with Gauss Principle adhering Gaussian Processes [7.643999306446022]
  We propose to combine insights from analytical mechanics with Gaussian process regression to improve the model's data efficiency and constraint integrity. Our model enables to infer the acceleration of the unconstrained system from data of the constrained system.
  arXiv  Detail & Related papers  (2020-04-23T15:26:51Z)
• On dissipative symplectic integration with applications to gradient-based optimization [77.34726150561087]
  We propose a geometric framework in which discretizations can be realized systematically. We show that a generalization of symplectic to nonconservative and in particular dissipative Hamiltonian systems is able to preserve rates of convergence up to a controlled error.
  arXiv  Detail & Related papers  (2020-04-15T00:36:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.