Element learning - a systematic approach of accelerating finite element-type methods via machine learning - applications to radiative transfer

Aug 5, 2026

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This paper presents element learning, a systematic machine-learning framework to accelerate finite element-type methods for the numerical solution of partial differential equations (PDEs). A neural network is trained element-wise to learn the PDE solution map from element geometry and local PDE parameters, producing operators for inter-element communication and element-wise solution recovery. After training, the model supports new domain geometries and parameter distributions without retraining, with simpler training than whole-domain approaches. Numerical experiments on radiative transfer show 5–10x speed-ups at a fixed 1e-3 relative L2 error.

LanguageEnglish

CategoryResearch & Report

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Introduction

Background and motivation

arXiv :2308 .02467v1 [math .NA] 4 Aug 2023 Element learning: a systematic approach of accelerating finite element-type methods via machine learning, with applications to radiative transfer Shukai Du∗1 and Samuel N. Stechmann 1,2 1 Department of Mathematics, University of Wisconsin–Madison 2 Department of Atmospheric and Oceanic Sciences, University of Wisconsin–Madison August 7, 2023 Abstract In this paper, we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations (PDEs) . The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion. This map takes input of the element geometry and the PDEs’parameters on that element, and gives output of two operators – (1) the in2out operator for inter-element communication, and (2) the in2sol operator (Green’s function) for element-wise solution recovery. A significant advantage of this approach is that, once trained, this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining. Also, the training is significantly simpler since it is done on the element level instead on the entire domain. We call this approach element learning. This method is closely related to hybridizbale discontinuous Galerkin (HDG) methods in the sense that the local solvers of HDG are replaced by machine learning approaches. Numerical tests are presented for an example PDE, the radiative transfer equation, in a variety of scenarios with idealized or realistic cloud fields, with smooth or sharp gradient in the cloud boundary transition. Under a fixed accuracy level of 10 ´3 in the relative L2 error, and polynomial degree p “ 6 in each element, we observe an approximately 5 to 10 times speed-up by element learning compared to a classical finite element-type method. Key words: scientific machine learning, spectral element, discontinuous Galerkin, hybridization, HDG, radiative transfer 1 Introduction 1.1 Background and motivation In the past decade,(artificial) neural networks and machine learning tools have surfaced as gamechanging technologies across numerous fields, resolving an array of challenging problems. Examples include image recognition [45, 44], playing the game go [67], protein folding [39], and large language models such as GPT3 [7] . ∗ Email: [sdu49@wisc.edu](sdu49@wisc.edu) Given these impressive results, it is reasonable to envision the potential of neural networks (NNs) for the numerical solution of partial differential equations (PDEs) or other scientific computing problems. There has been significant work in this direction [8, 57, 32, 25, 70, 60, 72, 42, 41, 10, 11, 27, 38, 74, 18, 24, 40, 66, 28, 36, 48, 56] . As examples, we mention two major groups –(1) neural networks as function approximators, (2) neural networks as operator approximators. The first type of these methods uses neural networks to approximate the solution of PDEs. Examples include Physics-Informed Neural Networks (PINNs) [61, 52, 50] and Deep Ritz methods [73] . These methods have demonstrated promising results, especially in the realm of highdimensional problems or inverse problems, since they seem to bypass the notorious ‘curse of dimensionality’, potentially attributable to the neural network’s efficient approximation capabilities in high-dimensional spaces [31, 30] . While the potential advantages are promising, it remains uncertain if the aforementioned methods have a real advantage over traditional algorithms, such as finite element-type methods, when addressing many classic PDEs. A crucial factor contributing to this uncertainty stems from the complex optimization problems and error landscapes—often non-convex—introduced by neural networks due to their hidden layers and non-linear activation functions. This complexity becomes particularly notable when considering th

FAQs

What is the core idea of element learning in this paper?

Element learning uses a neural network trained in an element-wise manner to learn the PDE solution map from each element’s geometry and local PDE parameters.

What operators does the trained network output?

It outputs two operators: an in2out operator for inter-element communication and an in2sol operator (Green’s function) for element-wise solution recovery.

How does the method behave after training?

Once trained, the network can solve PDEs for new domain geometries and parameter distributions without retraining.

Aug 5, 2026

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