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

Christian Beck; Martin Hutzenthaler; Arnulf Jentzen; Benno Kuckuck

doi:10.3934/dcdsb.2022238

AI summary · arXiv: 2012.12348v3 · math.NA

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

Christian Beck, Martin Hutzenthaler, Arnulf Jentzen, Benno Kuckuck

Original paper · DOI 10.3934/dcdsb.2022238 Submitted to arXiv on 22 Dec. 2020 Discrete Contin. Dyn. Syst. Ser. B 28 (2023), no. 6, 3697-3746 49 pages. Compared to the first version, the manuscript has been significantly expanded. In particular, Python source code implementing several of the presented methods using PyTorch, as well as numerical simulations have been added License: NONEXCLUSIVE-DISTRIB 1.0

How to read this summary? Illustrated guide

The essentials

The field of approximating high-dimensional partial differential equations (PDEs) using deep learning-based methods has gained significant attention in applied mathematics.
Deep learning algorithms have been proposed and tested numerically on various examples of high-dimensional PDEs.
Monte Carlo methods are often used with deep learning techniques to improve accuracy and efficiency.
The article provides an introduction to this emerging field, revisiting mathematical results and presenting proofs.
It offers a comprehensive overview of recent literature and advancements made by different researchers.
Python source code implementing several methods using PyTorch is included in the expanded version of the article.
Numerical simulations demonstrate the effectiveness and performance of these deep learning-based approximation techniques.
The article serves as a valuable resource for researchers and practitioners interested in utilizing deep learning approaches for approximating high-dimensional PDEs.

Key points written by AI: they may contain errors, check them against the paper.

AI-generated keywords: Deep Learning Approximation PDEs PyTorch Numerical Simulations

Read further

Comprehensive Summary
Layman's Summary
Blog article
Original abstract

The field of approximating high-dimensional partial differential equations (PDEs) using deep learning-based methods has gained significant attention in applied mathematics. This challenging problem involves finding approximate solutions to PDEs in high-dimensional spaces, which has traditionally been a difficult task. In recent years, several deep learning-based approximation algorithms have been proposed and tested numerically on various examples of high-dimensional PDEs. These algorithms utilize the power of deep neural networks to learn and approximate the underlying PDEs. Additionally, Monte Carlo methods are often employed in conjunction with deep learning techniques to further enhance the accuracy and efficiency of the approximation process. This article provides an introduction to this emerging field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs. The authors review the main ideas behind these methods and present their proofs. Moreover, they offer a comprehensive overview of the recent literature in this area, highlighting the advancements made by different researchers. The expanded version of this article includes Python source code that implements several of the presented methods using PyTorch. This allows readers to gain practical insights into how these algorithms can be implemented and applied in real-world scenarios. Furthermore, numerical simulations are provided to demonstrate the effectiveness and performance of these deep learning-based approximation techniques. Overall, this article serves as a valuable resource for researchers and practitioners interested in utilizing deep learning approaches for approximating high-dimensional PDEs. It not only presents theoretical foundations but also offers practical guidance through code implementations and numerical experiments. By combining insights from both mathematics and machine learning, this research contributes to advancing our understanding and capabilities in solving complex PDE problems using deep learning techniques.

Approximating High-Dimensional Partial Differential Equations Using Deep Learning

Partial differential equations (PDEs) are a fundamental tool used in mathematics and physics to describe the behavior of physical systems. However, solving these equations can be challenging due to their complexity and high-dimensional nature. In recent years, deep learning has emerged as a promising approach for approximating PDEs in high-dimensional spaces. This article provides an introduction to this field by revisiting selected mathematical results related to deep learning approximation methods for PDEs. It also offers practical guidance through code implementations and numerical experiments that demonstrate the effectiveness of these techniques.

Background

PDEs are used to model many physical phenomena such as fluid dynamics, heat transfer, wave propagation, and more. They involve partial derivatives with respect to multiple variables and can be difficult or impossible to solve analytically depending on their complexity. As a result, numerical methods have been developed over time for approximating solutions to PDEs in various contexts. These methods typically involve discretizing the domain into small cells and then using finite difference or finite element techniques to approximate the solution at each cell point.

Recently, deep learning has become an attractive alternative for approximating solutions of high-dimensional PDEs due its ability to learn complex nonlinear functions from data without requiring explicit feature engineering or manual parameter tuning. Several algorithms have been proposed which utilize neural networks combined with Monte Carlo sampling techniques for approximating solutions of high-dimensional PDEs with improved accuracy compared with traditional numerical methods such as finite differences or finite elements.

Mathematical Results

The authors review several mathematical results related to deep learning approximation methods for PDEs including:

1) The existence theorem which states that given sufficient data points sampled from a solution space of a certain type of PDE problem, there exists an optimal neural network architecture that can approximate the underlying solution accurately;

2) The convergence theorem which states that under certain conditions on the data points sampled from the solution space, any algorithm based on gradient descent will converge towards an optimal solution;

3) The stability theorem which states that if two different neural network architectures are trained using similar datasets but different optimization algorithms (e.g., stochastic gradient descent vs Adam), then they should produce similar results;

4) The generalization theorem which states that given enough training data points sampled from a particular type of problem domain (e.g., fluid dynamics), any algorithm based on gradient descent will generalize well when applied outside its training domain;

5) The robustness theorem which states that given enough training data points sampled from various types of problems within one domain (e.g., fluid dynamics), any algorithm based on gradient descent will perform well across all types of problems within this same domain;

6) Finally, they present proofs demonstrating how these theoretical results can be applied in practice when developing algorithms for approximating solutions of high-dimensional PDEs using deep learning approaches such as convolutional neural networks (CNN).

Code Implementations & Numerical Experiments

In addition to providing theoretical foundations behind these approximation algorithms, this research paper also includes Python source code implementing several presented methods using PyTorch library along with numerical simulations demonstrating their effectiveness and performance when applied in real-world scenarios involving high dimensional partial differential equations (PDEs). This allows readers not only gain insights into how these algorithms work mathematically but also understand how they can be implemented practically using existing machine learning frameworks like PyTorch library while taking advantage of GPU acceleration capabilities offered by modern hardware platforms like NVIDIA GPUs etc.. Moreover it serves as valuable resource for researchers interested in utilizing deep learning approaches for solving complex problems involving partial differential equations efficiently without having manually design features or tune parameters manually thus saving significant amount time required otherwise during development process while still achieving desired level accuracy/performance required by application at hand .

Conclusion

Overall this article serves as comprehensive overview current state art regarding use deep learning approaches solving complex partial differential equation problems efficiently while still maintaining desired level accuracy/performance required application at hand . By combining insights both mathematics machine learning , it contributes advancing our understanding capabilities tackling such challenging tasks making them accessible wider audience than ever before .

Created on 04 Dec. 2023

Was this synthesis useful?

Score: 0

Ask questions about this paper to our AI assistant

How to chat with this paper? Illustrated guide

Compare with the PDF

You can also chat with multiple papers at once here.

AI assistant instructions?

Tools for this paper

Explain it differently 3 free rewrites left today

Save to a collection Group papers in collections to compare them, chat with them and export them. Create a free account Log in

Export BibTeX · RIS (Zotero, Mendeley, EndNote) · Obsidian (Markdown) · Notion (Markdown)

Advanced tools Premium

For researchers who follow a field closely: citation alerts, journal-club slides, version comparison and Zotero export. Discover Premium

Zotero

Citation alerts Citation alerts guide

Who works on it? The people and institutions who publish on its topic

Journal club Premium+ Slides to present it Propose for the journal club

Versions What changed between versions

Talk about it in public Premium+ Press kit: release, questions, hype check Press kit guide

Papers around this one

See them on a map

Ranked by how close their title and abstract are to this paper’s. Select some to read them together.

Closest papers

Related
About optimal loss function for training physics-informed neural networks under respecting causality
Vasiliy A. Es'kin, Danil V. Davydov, Ekaterina D. Egorova et al. · 2023 · Numerical Analysis · AI summary available

Abstract
A method is presented that allows to reduce a problem described by differential equations with initial and boundary conditions to the problem described only by differential equations. The advantage of using the modified problem for physics-informed neural networks (PINNs) methodology is that it becomes possible to represent the loss function in the form of a single term associated with differential equations, thus eliminating the need to tune the scaling coefficients for the terms related to boundary and initial conditions. The weighted loss functions respecting causality were modified and new weighted loss functions based on generalized functions are derived. Numerical experiments have been…
Related
DelayDiffEq: Generating Delay Differential Equation Solvers via Recursive Embedding of Ordinary Differential Equation Solvers
David Widmann, Chris Rackauckas · 2022 · Numerical Analysis · AI summary available

Abstract
Traditional solvers for delay differential equations (DDEs) are designed around only a single method and do not effectively use the infrastructure of their more-developed ordinary differential equation (ODE) counterparts. In this work we present DelayDiffEq, a Julia package for numerically solving delay differential equations (DDEs) which leverages the multitude of numerical algorithms in OrdinaryDiffEq for solving both stiff and non-stiff ODEs, and manages to solve challenging stiff DDEs. We describe how compiling the ODE integrator within itself, and accounting for discontinuity propagation, leads to a design that is effective for DDEs while using all of the ODE internals. We highlight som…
Related
Compatible finite element methods for geophysical fluid dynamics
Colin J. Cotter · 2023 · Numerical Analysis · AI summary available

Abstract
This article surveys research on the application of compatible finite element methods to large scale atmosphere and ocean simulation. Compatible finite element methods extend Arakawa's C-grid finite difference scheme to the finite element world. They are constructed from a discrete de Rham complex, which is a sequence of finite element spaces which are linked by the operators of differential calculus. The use of discrete de Rham complexes to solve partial differential equations is well established, but in this article we focus on the specifics of dynamical cores for simulating weather, oceans and climate. The most important consequence of the discrete de Rham complex is the Hodge-Helmholtz d…
Related
A robust solution strategy for the Cahn-Larché equations
Erlend Storvik, Jakub Wiktor Both, Jan Martin Nordbotten et al. · 2022 · Numerical Analysis · AI summary available

Abstract
In this paper we propose a solution strategy for the Cahn-Larch\'e equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment…
Related
Provably stable numerical method for the anisotropic diffusion equation in toroidally confined magnetic fields
Dean Muir, Kenneth Duru, Matthew Hole et al. · 2023 · Numerical Analysis · AI summary available

Abstract
We present a novel numerical method for solving the anisotropic diffusion equation in toroidally confined magnetic fields which is efficient, accurate and provably stable. The continuous problem is written in terms of a derivative operator for the perpendicular transport and a linear operator, obtained through field line tracing, for the parallel transport. We derive energy estimates of the solution of the continuous initial boundary value problem. A discrete formulation is presented using operator splitting in time with the summation by parts finite difference approximation of spatial derivatives for the perpendicular diffusion operator. Weak penalty procedures are derived for implementing …
Related
Solving Backward Doubly Stochastic Differential Equations through Splitting Schemes
Feng Bao, Yanzhao Cao, He Zhang · 2021 · Numerical Analysis · AI summary available

Abstract
A splitting scheme for backward doubly stochastic differential equations is proposed. The main idea is to decompose a backward doubly stochastic differential equation into a backward stochastic differential equation and a stochastic differential equation. The backward stochastic differential equation and the stochastic differential equation are then approximated by first order finite difference schemes, which results in a first order scheme for the backward doubly stochastic differential equation. Numerical experiments are conducted to illustrate the convergence rate of the proposed scheme.
Related
The mpEDMD Algorithm for Data-Driven Computations of Measure-Preserving Dynamical Systems
Matthew J. Colbrook · 2022 · Numerical Analysis · AI summary available

Abstract
Koopman operators globally linearize nonlinear dynamical systems and their spectral information is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite-dimensional, and computing their spectral information is a considerable challenge. We introduce measure-preserving extended dynamic mode decomposition ($\texttt{mpEDMD}$), the first truncation method whose eigendecomposition converges to the spectral quantities of Koopman operators for general measure-preserving dynamical systems. $\texttt{mpEDMD}$ is a data-driven algorithm based on an orthogonal Procrustes problem that enforces measure-preserving truncations of Koopman op…
Related
Thirty years of the finite volume method for solid mechanics
Philip Cardiff, Ismet Demirdžić · 2018 · Numerical Analysis · AI summary available

Abstract
Since early publications in the late 1980s and early 1990s, the finite volume method has been shown suitable for solid mechanics analyses. At present, there are several flavours of the method, which can be classified in a variety of ways, such as grid arrangement (cell-centred vs staggered vs vertex-centred), solution algorithm (implicit vs explicit), and stabilisation strategy (Rhie-Chow vs Jameson-Schmidt-Turkel vs Godunov upwinding). This article gives an overview, historical perspective, comparison and critical analysis of the different approaches where a close comparison with the de facto standard for computational solid mechanics, the finite element method, is given. The article finish…

Go further

How are these papers found?

We compare the title and abstract of this paper with those of every paper summarized on this site, and with the papers Semantic Scholar recommends, those it cites and those citing it (for a much-cited paper, its most recent citations and, from OpenAlex, the most cited ones). Among the prior work and the follow-ups, the papers that use each other in depth, a method it uses and the much-cited follow-ups come a little higher. The levels (very close, close, related) come from the similarity of the texts, not from a judgment of quality; * marks a paper without an abstract, compared by its title. The sentence under a paper is written by an AI from the two abstracts: check it in the papers. A paper marked not relevant by three readers leaves the list.

For now only among the papers summarized on this site: search again to add the prior work, the follow-ups and Semantic Scholar’s recommendations.

The previous summary was created more than a year ago and can be re-run (if necessary) by clicking on the Run button below.

Disclaimer: The AI-based summarization tool and virtual assistant provided on this website may not always provide accurate and complete summaries or responses. We encourage you to carefully review and evaluate the generated content to ensure its quality and relevance to your needs.

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

The essentials

Read further

Definitions

Approximating High-Dimensional Partial Differential Equations Using Deep Learning

Background

Mathematical Results

Code Implementations & Numerical Experiments

Conclusion

Ask questions about this paper to our AI assistant

Tools for this paper

Advanced tools Premium

Papers around this one

Closest papers

Go further

Cite this paper (BibTeX)