Nonlinear embeddings for conserving Hamiltonians and other quantities with Neural Galerkin schemes

• URL: http://arxiv.org/abs/2310.07485v1
• Date: Wed, 11 Oct 2023 13:32:04 GMT
• Title: Nonlinear embeddings for conserving Hamiltonians and other quantities with Neural Galerkin schemes
• Authors: Paul Schwerdtner, Philipp Schulze, Jules Berman, Benjamin Peherstorfer
• Abstract summary: This work focuses on the conservation of quantities such as Hamiltonians, mass, and momentum when solution fields of partial differential equations are approximated with nonlinear parametrizations such as deep networks. The proposed approach builds on Neural Galerkin schemes that are based on the Dirac--Frenkel variational principle to train nonlinear parametrizations sequentially in time.
• Score: 1.1509084774278489
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This work focuses on the conservation of quantities such as Hamiltonians, mass, and momentum when solution fields of partial differential equations are approximated with nonlinear parametrizations such as deep networks. The proposed approach builds on Neural Galerkin schemes that are based on the Dirac--Frenkel variational principle to train nonlinear parametrizations sequentially in time. We first show that only adding constraints that aim to conserve quantities in continuous time can be insufficient because the nonlinear dependence on the parameters implies that even quantities that are linear in the solution fields become nonlinear in the parameters and thus are challenging to discretize in time. Instead, we propose Neural Galerkin schemes that compute at each time step an explicit embedding onto the manifold of nonlinearly parametrized solution fields to guarantee conservation of quantities. The embeddings can be combined with standard explicit and implicit time integration schemes. Numerical experiments demonstrate that the proposed approach conserves quantities up to machine precision.

Related papers

• Numerical solution of nonlinear Schrödinger equation by a hybrid pseudospectral-variational quantum algorithm [0.0]
  Time-dependent one-dimensional nonlinear Schr"odinger equation (NLSE) is solved numerically by a hybrid pseudospectral-variational quantum algorithm. We analyze the accuracy of the quantum algorithm and compare it with classical approaches.
  arXiv  Detail & Related papers  (2024-07-03T10:40:31Z)
• Neural empirical interpolation method for nonlinear model reduction [0.0]
  We introduce the neural empirical method (NEIM) for reducing the time complexity of computing the nonlinear term in a reduced order model (ROM) NEIM is a greedy algorithm which accomplishes this reduction by approximating an affine decomposition of the nonlinear term of the ROM. Because NEIM is based on a greedy strategy, we are able to provide a basic error analysis to investigate its performance.
  arXiv  Detail & Related papers  (2024-06-05T18:17:33Z)
• The Challenges of the Nonlinear Regime for Physics-Informed Neural Networks [0.0]
  We show how the NTK perspective falls short in the nonlinear scenario. We explore the convergence guarantees of such methods in both linear and nonlinear cases.
  arXiv  Detail & Related papers  (2024-02-06T10:24:36Z)
• Estimation Sample Complexity of a Class of Nonlinear Continuous-time Systems [0.0]
  We present a method of parameter estimation for large class of nonlinear systems, namely those in which the state consists of output derivatives and the flow is linear in the parameter. The method, which solves for the unknown parameter by directly inverting the dynamics using regularized linear regression, is based on new design and analysis ideas for differentiation filtering and regularized least squares.
  arXiv  Detail & Related papers  (2023-12-08T21:42:11Z)
• Non-Parametric Learning of Stochastic Differential Equations with Non-asymptotic Fast Rates of Convergence [65.63201894457404]
  We propose a novel non-parametric learning paradigm for the identification of drift and diffusion coefficients of non-linear differential equations. The key idea essentially consists of fitting a RKHS-based approximation of the corresponding Fokker-Planck equation to such observations.
  arXiv  Detail & Related papers  (2023-05-24T20:43:47Z)
• Dynamical chaos in nonlinear Schr\"odinger models with subquadratic power nonlinearity [137.6408511310322]
  We deal with a class of nonlinear Schr"odinger lattices with random potential and subquadratic power nonlinearity. We show that the spreading process is subdiffusive and has complex microscopic organization. The limit of quadratic power nonlinearity is also discussed and shown to result in a delocalization border.
  arXiv  Detail & Related papers  (2023-01-20T16:45:36Z)
• Structure-Preserving Learning Using Gaussian Processes and Variational Integrators [62.31425348954686]
  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.
  arXiv  Detail & Related papers  (2021-12-10T11:09:29Z)
• Optimization on manifolds: A symplectic approach [127.54402681305629]
  We propose a dissipative extension of Dirac's theory of constrained Hamiltonian systems as a general framework for solving optimization problems. Our class of (accelerated) algorithms are not only simple and efficient but also applicable to a broad range of contexts.
  arXiv  Detail & Related papers  (2021-07-23T13:43:34Z)
• The Connection between Discrete- and Continuous-Time Descriptions of Gaussian Continuous Processes [60.35125735474386]
  We show that discretizations yielding consistent estimators have the property of invariance under coarse-graining' This result explains why combining differencing schemes for derivatives reconstruction and local-in-time inference approaches does not work for time series analysis of second or higher order differential equations.
  arXiv  Detail & Related papers  (2021-01-16T17:11:02Z)
• Coarse-Grained Nonlinear System Identification [0.0]
  We introduce Coarse-Grained Dynamics, an efficient and universal parameterization of nonlinear system dynamics based on the Volterra series expansion. We demonstrate the properties of this approach on a simple synthetic problem. We also demonstrate this approach experimentally, showing that it identifies an accurate model of the nonlinear voltage to dynamics of a tungsten filament with less than a second of experimental data.
  arXiv  Detail & Related papers  (2020-10-14T06:45:51Z)
• 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.