An overview on deep learning-based approximation methods for partial differential equations

• URL: http://arxiv.org/abs/2012.12348v2
• Date: Fri, 5 Mar 2021 10:53:55 GMT
• Title: An overview on deep learning-based approximation methods for partial differential equations
• Authors: Christian Beck, Martin Hutzenthaler, Arnulf Jentzen, Benno Kuckuck
• Abstract summary: It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs) Several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs.
• Score: 2.4087148947930634
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research, we review some of the main ideas of deep learning-based approximation methods for PDEs, we revisit one of the central mathematical results for deep neural network approximations for PDEs, and we provide an overview of the recent literature in this area of research.

Related papers

• An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning [5.75055574132362]
  approximation of solutions of partial differential equations with numerical algorithms is a central topic in applied mathematics. One class of methods which has received a lot of attention in recent years are machine learning-based methods. This article aims to provide an introduction to some of these methods and the mathematical theory on which they are based.
  arXiv  Detail & Related papers  (2024-08-23T16:57:34Z)
• 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)
• Optimizing Solution-Samplers for Combinatorial Problems: The Landscape of Policy-Gradient Methods [52.0617030129699]
  We introduce a novel theoretical framework for analyzing the effectiveness of DeepMatching Networks and Reinforcement Learning methods. Our main contribution holds for a broad class of problems including Max-and Min-Cut, Max-$k$-Bipartite-Bi, Maximum-Weight-Bipartite-Bi, and Traveling Salesman Problem. As a byproduct of our analysis we introduce a novel regularization process over vanilla descent and provide theoretical and experimental evidence that it helps address vanishing-gradient issues and escape bad stationary points.
  arXiv  Detail & Related papers  (2023-10-08T23:39:38Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• Partial Differential Equations Meet Deep Neural Networks: A Survey [10.817323756266527]
  Problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics such as computational fluid dynamics. Deep Neural Networks (DNNs) for PDEs have emerged as an effective means of solving PDEs.
  arXiv  Detail & Related papers  (2022-10-27T07:01:56Z)
• Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning [6.1678491628787455]
  Combination of Monte Carlo methods and deep learning has led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations based on associated differential equations (SDEs) It is therefore crucial to rely on adequate gradient estimators that exhibit low variance in order to reach convergence accurately and swiftly.
  arXiv  Detail & Related papers  (2022-06-21T17:59:39Z)
• Finite Expression Method for Solving High-Dimensional Partial Differential Equations [5.736353542430439]
  This paper introduces a new methodology that seeks an approximate PDE solution in the space of functions with finitely many analytic expressions. It is proved in approximation theory that FEX can avoid the curse of dimensionality. An approximate solution with finite analytic expressions also provides interpretable insights into the ground truth PDE solution.
  arXiv  Detail & Related papers  (2022-06-21T05:51:10Z)
• Model-Based Deep Learning: On the Intersection of Deep Learning and Optimization [101.32332941117271]
  Decision making algorithms are used in a multitude of different applications. Deep learning approaches that use highly parametric architectures tuned from data without relying on mathematical models are becoming increasingly popular. Model-based optimization and data-centric deep learning are often considered to be distinct disciplines.
  arXiv  Detail & Related papers  (2022-05-05T13:40:08Z)
• Lie Point Symmetry Data Augmentation for Neural PDE Solvers [69.72427135610106]
  We present a method, which can partially alleviate this problem, by improving neural PDE solver sample complexity. In the context of PDEs, it turns out that we are able to quantitatively derive an exhaustive list of data transformations. We show how it can easily be deployed to improve neural PDE solver sample complexity by an order of magnitude.
  arXiv  Detail & Related papers  (2022-02-15T18:43:17Z)
• Adversarial Multi-task Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations [9.823102211212582]
  We introduce the novel approach of employing multi-task learning techniques, the uncertainty-weighting loss and the gradients surgery, in the context of learning PDE solutions. In the experiments, our proposed methods are found to be effective and reduce the error on the unseen data points as compared to the previous approaches.
  arXiv  Detail & Related papers  (2021-04-29T13:17:46Z)
• A Survey on Deep Semi-supervised Learning [51.26862262550445]
  We first present a taxonomy for deep semi-supervised learning that categorizes existing methods. We then offer a detailed comparison of these methods in terms of the type of losses, contributions, and architecture differences.
  arXiv  Detail & Related papers  (2021-02-28T16:22:58Z)

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.