Solving non-linear Kolmogorov equations in large dimensions by using deep learning: a numerical comparison of discretization schemes

• URL: http://arxiv.org/abs/2012.07747v2
• Date: Mon, 28 Dec 2020 13:41:35 GMT
• Title: Solving non-linear Kolmogorov equations in large dimensions by using deep learning: a numerical comparison of discretization schemes
• Authors: Nicolas Macris and Raffaele Marino
• Abstract summary: Non-linear partial differential Kolmogorov equations are successfully used to describe a wide range of time dependent phenomena. Deep learning has been introduced to solve these equations in high-dimensional regimes. We show that, for some discretization schemes, improvements in the accuracy are possible without affecting the observed computational complexity.
• Score: 16.067228939231047
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Non-linear partial differential Kolmogorov equations are successfully used to describe a wide range of time dependent phenomena, in natural sciences, engineering or even finance. For example, in physical systems, the Allen-Cahn equation describes pattern formation associated to phase transitions. In finance, instead, the Black-Scholes equation describes the evolution of the price of derivative investment instruments. Such modern applications often require to solve these equations in high-dimensional regimes in which classical approaches are ineffective. Recently, an interesting new approach based on deep learning has been introduced by E, Han, and Jentzen [1][2]. The main idea is to construct a deep network which is trained from the samples of discrete stochastic differential equations underlying Kolmogorov's equation. The network is able to approximate, numerically at least, the solutions of the Kolmogorov equation with polynomial complexity in whole spatial domains. In this contribution we study variants of the deep networks by using different discretizations schemes of the stochastic differential equation. We compare the performance of the associated networks, on benchmarked examples, and show that, for some discretization schemes, improvements in the accuracy are possible without affecting the observed computational complexity.

Related papers

• Solving Fractional Differential Equations on a Quantum Computer: A Variational Approach [0.1492582382799606]
  We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation cost scales economically with the number of time steps.
  arXiv  Detail & Related papers  (2024-06-13T02:27:16Z)
• Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
  arXiv  Detail & Related papers  (2024-06-05T17:59:22Z)
• Physics-Informed Generator-Encoder Adversarial Networks with Latent Space Matching for Stochastic Differential Equations [14.999611448900822]
  We propose a new class of physics-informed neural networks to address the challenges posed by forward, inverse, and mixed problems in differential equations. Our model consists of two key components: the generator and the encoder, both updated alternately by gradient descent. In contrast to previous approaches, we employ an indirect matching that operates within the lower-dimensional latent feature space.
  arXiv  Detail & Related papers  (2023-11-03T04:29:49Z)
• deepFDEnet: A Novel Neural Network Architecture for Solving Fractional Differential Equations [0.0]
  In each fractional differential equation, a deep neural network is used to approximate the unknown function. The results show that the proposed architecture solves different forms of fractional differential equations with excellent precision.
  arXiv  Detail & Related papers  (2023-09-14T12:58:40Z)
• Correspondence between open bosonic systems and stochastic differential equations [77.34726150561087]
  We show that there can also be an exact correspondence at finite $n$ when the bosonic system is generalized to include interactions with the environment. A particular system with the form of a discrete nonlinear Schr"odinger equation is analyzed in more detail.
  arXiv  Detail & Related papers  (2023-02-03T19:17:37Z)
• Tunable Complexity Benchmarks for Evaluating Physics-Informed Neural Networks on Coupled Ordinary Differential Equations [64.78260098263489]
  In this work, we assess the ability of physics-informed neural networks (PINNs) to solve increasingly-complex coupled ordinary differential equations (ODEs) We show that PINNs eventually fail to produce correct solutions to these benchmarks as their complexity increases. We identify several reasons why this may be the case, including insufficient network capacity, poor conditioning of the ODEs, and high local curvature, as measured by the Laplacian of the PINN loss.
  arXiv  Detail & Related papers  (2022-10-14T15:01:32Z)
• Semi-supervised Learning of Partial Differential Operators and Dynamical Flows [68.77595310155365]
  We present a novel method that combines a hyper-network solver with a Fourier Neural Operator architecture. We test our method on various time evolution PDEs, including nonlinear fluid flows in one, two, and three spatial dimensions. The results show that the new method improves the learning accuracy at the time point of supervision point, and is able to interpolate and the solutions to any intermediate time.
  arXiv  Detail & Related papers  (2022-07-28T19:59:14Z)
• D-CIPHER: Discovery of Closed-form Partial Differential Equations [80.46395274587098]
  We propose D-CIPHER, which is robust to measurement artifacts and can uncover a new and very general class of differential equations. We further design a novel optimization procedure, CoLLie, to help D-CIPHER search through this class efficiently.
  arXiv  Detail & Related papers  (2022-06-21T17:59:20Z)
• Continuous Convolutional Neural Networks: Coupled Neural PDE and ODE [1.1897857181479061]
  This work proposes a variant of Convolutional Neural Networks (CNNs) that can learn the hidden dynamics of a physical system. Instead of considering the physical system such as image, time -series as a system of multiple layers, this new technique can model a system in the form of Differential Equation (DEs)
  arXiv  Detail & Related papers  (2021-10-30T21:45:00Z)
• 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)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)

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.