Unsupervised Reservoir Computing for Solving Ordinary Differential Equations

• URL: http://arxiv.org/abs/2108.11417v1
• Date: Wed, 25 Aug 2021 18:16:42 GMT
• Title: Unsupervised Reservoir Computing for Solving Ordinary Differential Equations
• Authors: Marios Mattheakis, Hayden Joy, Pavlos Protopapas
• Abstract summary: unsupervised reservoir computing (RC), an echo-state recurrent neural network capable of discovering approximate solutions that satisfy ordinary differential equations (ODEs) We use Bayesian optimization to efficiently discover optimal sets in a high-dimensional hyper- parameter space and numerically show that one set is robust and can be used to solve an ODE for different initial conditions and time ranges.
• Score: 1.6371837018687636
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: There is a wave of interest in using unsupervised neural networks for solving differential equations. The existing methods are based on feed-forward networks, {while} recurrent neural network differential equation solvers have not yet been reported. We introduce an unsupervised reservoir computing (RC), an echo-state recurrent neural network capable of discovering approximate solutions that satisfy ordinary differential equations (ODEs). We suggest an approach to calculate time derivatives of recurrent neural network outputs without using backpropagation. The internal weights of an RC are fixed, while only a linear output layer is trained, yielding efficient training. However, RC performance strongly depends on finding the optimal hyper-parameters, which is a computationally expensive process. We use Bayesian optimization to efficiently discover optimal sets in a high-dimensional hyper-parameter space and numerically show that one set is robust and can be used to solve an ODE for different initial conditions and time ranges. A closed-form formula for the optimal output weights is derived to solve first order linear equations in a backpropagation-free learning process. We extend the RC approach by solving nonlinear system of ODEs using a hybrid optimization method consisting of gradient descent and Bayesian optimization. Evaluation of linear and nonlinear systems of equations demonstrates the efficiency of the RC ODE solver.

Related papers

• Linearization of ReLU Activation Function for Neural Network-Embedded Optimization:Optimal Day-Ahead Energy Scheduling [0.2900810893770134]
  In some applications such as battery degradation neural network-based microgrid day-ahead energy scheduling, the input features of the trained learning model are variables to be solved in optimization models. The use of nonlinear activation functions in the neural network will make such problems extremely hard to solve if not unsolvable. This paper investigated different methods for linearizing the nonlinear activation functions with a particular focus on the widely used rectified linear unit (ReLU) function.
  arXiv  Detail & Related papers  (2023-10-03T02:47:38Z)
• An Optimization-based Deep Equilibrium Model for Hyperspectral Image Deconvolution with Convergence Guarantees [71.57324258813675]
  We propose a novel methodology for addressing the hyperspectral image deconvolution problem. A new optimization problem is formulated, leveraging a learnable regularizer in the form of a neural network. The derived iterative solver is then expressed as a fixed-point calculation problem within the Deep Equilibrium framework.
  arXiv  Detail & Related papers  (2023-06-10T08:25:16Z)
• Constrained Optimization via Exact Augmented Lagrangian and Randomized Iterative Sketching [55.28394191394675]
  We develop an adaptive inexact Newton method for equality-constrained nonlinear, nonIBS optimization problems. We demonstrate the superior performance of our method on benchmark nonlinear problems, constrained logistic regression with data from LVM, and a PDE-constrained problem.
  arXiv  Detail & Related papers  (2023-05-28T06:33:37Z)
• OKRidge: Scalable Optimal k-Sparse Ridge Regression [21.17964202317435]
  We propose a fast algorithm, OKRidge, for sparse ridge regression. We also propose a method to warm-start our solver, which leverages a beam search.
  arXiv  Detail & Related papers  (2023-04-13T17:34:44Z)
• Neural Basis Functions for Accelerating Solutions to High Mach Euler Equations [63.8376359764052]
  We propose an approach to solving partial differential equations (PDEs) using a set of neural networks. We regress a set of neural networks onto a reduced order Proper Orthogonal Decomposition (POD) basis. These networks are then used in combination with a branch network that ingests the parameters of the prescribed PDE to compute a reduced order approximation to the PDE.
  arXiv  Detail & Related papers  (2022-08-02T18:27:13Z)
• Meta-Solver for Neural Ordinary Differential Equations [77.8918415523446]
  We investigate how the variability in solvers' space can improve neural ODEs performance. We show that the right choice of solver parameterization can significantly affect neural ODEs models in terms of robustness to adversarial attacks.
  arXiv  Detail & Related papers  (2021-03-15T17:26:34Z)
• Deep neural network for solving differential equations motivated by Legendre-Galerkin approximation [16.64525769134209]
  We explore the performance and accuracy of various neural architectures on both linear and nonlinear differential equations. We implement a novel Legendre-Galerkin Deep Neural Network (LGNet) algorithm to predict solutions to differential equations.
  arXiv  Detail & Related papers  (2020-10-24T20:25:09Z)
• Accelerating Feedforward Computation via Parallel Nonlinear Equation Solving [106.63673243937492]
  Feedforward computation, such as evaluating a neural network or sampling from an autoregressive model, is ubiquitous in machine learning. We frame the task of feedforward computation as solving a system of nonlinear equations. We then propose to find the solution using a Jacobi or Gauss-Seidel fixed-point method, as well as hybrid methods of both. Our method is guaranteed to give exactly the same values as the original feedforward computation with a reduced (or equal) number of parallelizable iterations, and hence reduced time given sufficient parallel computing power.
  arXiv  Detail & Related papers  (2020-02-10T10:11:31Z)
• Enhancement of shock-capturing methods via machine learning [0.0]
  We develop an improved finite-volume method for simulating PDEs with discontinuous solutions. We train a neural network to improve the results of a fifth-order WENO method. We find that our method outperforms WENO in simulations where the numerical solution becomes overly diffused.
  arXiv  Detail & Related papers  (2020-02-06T21:51:39Z)
• Channel Assignment in Uplink Wireless Communication using Machine Learning Approach [54.012791474906514]
  This letter investigates a channel assignment problem in uplink wireless communication systems. Our goal is to maximize the sum rate of all users subject to integer channel assignment constraints. Due to high computational complexity, machine learning approaches are employed to obtain computational efficient solutions.
  arXiv  Detail & Related papers  (2020-01-12T15:54:20Z)

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.