Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control

• URL: http://arxiv.org/abs/2105.14094v1
• Date: Fri, 28 May 2021 20:25:40 GMT
• Title: Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control
• Authors: Mark Ainsworth and Justin Dong
• Abstract summary: We present a new approach to using neural networks to approximate the solutions of variational equations. We use a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence of neural networks.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: We present a new approach to using neural networks to approximate the solutions of variational equations, based on the adaptive construction of a sequence of finite-dimensional subspaces whose basis functions are realizations of a sequence of neural networks. The finite-dimensional subspaces are then used to define a standard Galerkin approximation of the variational equation. This approach enjoys a number of advantages, including: the sequential nature of the algorithm offers a systematic approach to enhancing the accuracy of a given approximation; the sequential enhancements provide a useful indicator for the error that can be used as a criterion for terminating the sequential updates; the basic approach is largely oblivious to the nature of the partial differential equation under consideration; and, some basic theoretical results are presented regarding the convergence (or otherwise) of the method which are used to formulate basic guidelines for applying the method.

Related papers

• Augmented neural forms with parametric boundary-matching operators for solving ordinary differential equations [0.0]
  This paper introduces a formalism for systematically crafting proper neural forms with boundary matches that are amenable to optimization. It describes a novel technique for converting problems with Neumann or Robin conditions into equivalent problems with parametric Dirichlet conditions. The proposed augmented neural forms approach was tested on a set of diverse problems, encompassing first- and second-order ordinary differential equations, as well as first-order systems.
  arXiv  Detail & Related papers  (2024-04-30T11:10:34Z)
• A new approach to generalisation error of machine learning algorithms: Estimates and convergence [0.0]
  We introduce a new approach to the estimation of the (generalisation) error and to convergence. Our results include estimates of the error without any structural assumption on the neural networks.
  arXiv  Detail & Related papers  (2023-06-23T20:57:31Z)
• A Recursively Recurrent Neural Network (R2N2) Architecture for Learning Iterative Algorithms [64.3064050603721]
  We generalize Runge-Kutta neural network to a recurrent neural network (R2N2) superstructure for the design of customized iterative algorithms. We demonstrate that regular training of the weight parameters inside the proposed superstructure on input/output data of various computational problem classes yields similar iterations to Krylov solvers for linear equation systems, Newton-Krylov solvers for nonlinear equation systems, and Runge-Kutta solvers for ordinary differential equations.
  arXiv  Detail & Related papers  (2022-11-22T16:30:33Z)
• Scalable computation of prediction intervals for neural networks via matrix sketching [79.44177623781043]
  Existing algorithms for uncertainty estimation require modifying the model architecture and training procedure. This work proposes a new algorithm that can be applied to a given trained neural network and produces approximate prediction intervals.
  arXiv  Detail & Related papers  (2022-05-06T13:18:31Z)
• An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations [68.8204255655161]
  Filtering equations play a central role in many real-life applications, including numerical weather prediction, finance and engineering. One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method. We combine this method with a neural network representation to produce an approximation of the unnormalised conditional distribution of the signal process.
  arXiv  Detail & Related papers  (2022-01-10T11:01:36Z)
• Numerical Solution of Stiff Ordinary Differential Equations with Random Projection Neural Networks [0.0]
  We propose a numerical scheme based on Random Projection Neural Networks (RPNN) for the solution of Ordinary Differential Equations (ODEs) We show that our proposed scheme yields good numerical approximation accuracy without being affected by the stiffness, thus outperforming in same cases the textttode45 and textttode15s functions.
  arXiv  Detail & Related papers  (2021-08-03T15:49:17Z)
• Fractal Structure and Generalization Properties of Stochastic Optimization Algorithms [71.62575565990502]
  We prove that the generalization error of an optimization algorithm can be bounded on the complexity' of the fractal structure that underlies its generalization measure. We further specialize our results to specific problems (e.g., linear/logistic regression, one hidden/layered neural networks) and algorithms.
  arXiv  Detail & Related papers  (2021-06-09T08:05:36Z)
• Optimal oracle inequalities for solving projected fixed-point equations [53.31620399640334]
  We study methods that use a collection of random observations to compute approximate solutions by searching over a known low-dimensional subspace of the Hilbert space. We show how our results precisely characterize the error of a class of temporal difference learning methods for the policy evaluation problem with linear function approximation.
  arXiv  Detail & Related papers  (2020-12-09T20:19:32Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• Distributed Value Function Approximation for Collaborative Multi-Agent Reinforcement Learning [2.7071541526963805]
  We propose several novel distributed gradient-based temporal difference algorithms for multi-agent off-policy learning. The proposed algorithms differ by their form, definition of eligibility traces, selection of time scales and the way of incorporating consensus iterations. It is demonstrated how the adopted methodology can be applied to temporal-difference algorithms under weaker information structure constraints.
  arXiv  Detail & Related papers  (2020-06-18T11:46:09Z)
• Model Reduction and Neural Networks for Parametric PDEs [9.405458160620533]
  We develop a framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology.
  arXiv  Detail & Related papers  (2020-05-07T00:09:27Z)

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.