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In this paper, inspired by the multigrid method, we propose a multi-level deep framework for deep solvers. Overall, it divides the entire training process into different levels of training. At each level of training, an adaptive sampling…

Numerical Analysis · Mathematics 2026-02-23 Yu Yang , Qiaolin He

Designing universal artificial intelligence (AI) solver for partial differential equations (PDEs) is an open-ended problem and a significant challenge in science and engineering. Currently, data-driven solvers have achieved great success,…

Machine Learning · Computer Science 2025-02-24 Qinglong Ma , Peizhi Zhao , Sen Wang , Tao Song

Solving partial differential equations (PDEs) by numerical methods meet computational cost challenge for getting the accurate solution since fine grids and small time steps are required. Machine learning can accelerate this process, but…

Numerical Analysis · Mathematics 2025-01-28 Qi Wang , Yuan Mi , Haoyun Wang , Yi Zhang , Ruizhi Chengze , Hongsheng Liu , Ji-Rong Wen , Hao Sun

The recent rise of deep learning has led to numerous applications, including solving partial differential equations using Physics-Informed Neural Networks. This approach has proven highly effective in several academic cases. However, their…

Numerical Analysis · Mathematics 2024-10-07 Marien Chenaud , Frédéric Magoulès , José Alves

Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…

Machine Learning · Computer Science 2019-11-22 Jonathan B. Freund , Jonathan F. MacArt , Justin Sirignano

Across numerous applications, forecasting relies on numerical solvers for partial differential equations (PDEs). Although the use of deep-learning techniques has been proposed, actual applications have been restricted by the fact the…

Machine Learning · Computer Science 2020-01-28 Philipp Haehnel , Jakub Marecek , Julien Monteil , Fearghal O'Donncha

We investigate numerous structural connections between numerical algorithms for partial differential equations (PDEs) and neural architectures. Our goal is to transfer the rich set of mathematical foundations from the world of PDEs to…

Numerical Analysis · Mathematics 2022-03-22 Tobias Alt , Karl Schrader , Matthias Augustin , Pascal Peter , Joachim Weickert

Neural networks have shown promising potential in accelerating the numerical simulation of systems governed by partial differential equations (PDEs). Different from many existing neural network surrogates operating on high-dimensional…

Machine Learning · Computer Science 2025-01-09 Zijie Li , Saurabh Patil , Francis Ogoke , Dule Shu , Wilson Zhen , Michael Schneier , John R. Buchanan, , Amir Barati Farimani

Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…

Numerical Analysis · Mathematics 2025-03-10 Mingxing Weng , Zhiping Mao , Jie Shen

As the artificial intelligence community advances into the era of large models with billions of parameters, distributed training and inference have become essential. While various parallelism strategies-data, model, sequence, and…

Machine Learning · Computer Science 2025-03-13 Ruifeng She , Bowen Pang , Kai Li , Zehua Liu , Tao Zhong

Deep neural networks have established themselves as one of the most promising machine learning techniques. Training such models at large scales is often parallelized, giving rise to the concept of distributed deep learning. Distributed…

Quantum Physics · Physics 2022-11-15 Lirandë Pira , Chris Ferrie

Most research on novel techniques for 3D Medical Image Segmentation (MIS) is currently done using Deep Learning with GPU accelerators. The principal challenge of such technique is that a single input can easily cope computing resources, and…

Machine Learning · Computer Science 2021-11-01 Josep Lluis Berral , Oriol Aranda , Juan Luis Dominguez , Jordi Torres

Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting…

Computational Physics · Physics 2021-01-06 Kiwon Um , Robert Brand , Yun , Fei , Philipp Holl , Nils Thuerey

(Partial) differential equations (PDEs) are fundamental tools for describing natural phenomena, making their solution crucial in science and engineering. While traditional methods, such as the finite element method, provide reliable…

Machine Learning · Computer Science 2025-03-11 Viggo Moro , Luiz F. O. Chamon

We propose a neural network-based meta-learning method to efficiently solve partial differential equation (PDE) problems. The proposed method is designed to meta-learn how to solve a wide variety of PDE problems, and uses the knowledge for…

Machine Learning · Statistics 2023-10-23 Tomoharu Iwata , Yusuke Tanaka , Naonori Ueda

Training machine learning models in parallel is an increasingly important workload. We accelerate distributed parallel training by designing a communication primitive that uses a programmable switch dataplane to execute a key step of the…

Distributed, Parallel, and Cluster Computing · Computer Science 2020-10-01 Amedeo Sapio , Marco Canini , Chen-Yu Ho , Jacob Nelson , Panos Kalnis , Changhoon Kim , Arvind Krishnamurthy , Masoud Moshref , Dan R. K. Ports , Peter Richtárik

This work proposes an autoencoder neural network as a non-linear generalization of projection-based methods for solving Partial Differential Equations (PDEs). The proposed deep learning architecture presented is capable of generating the…

Computational Physics · Physics 2020-06-25 Jaime Lopez Garcia , Angel Rivero Jimenez

Partial differential equations (PDEs) are crucial in modeling diverse phenomena across scientific disciplines, including seismic and medical imaging, computational fluid dynamics, image processing, and neural networks. Solving these PDEs at…

Distributed, Parallel, and Cluster Computing · Computer Science 2025-01-07 George Bisbas , Rhodri Nelson , Mathias Louboutin , Fabio Luporini , Paul H. J. Kelly , Gerard Gorman

The increasingly deeper neural networks hinder the democratization of privacy-enhancing distributed learning, such as federated learning (FL), to resource-constrained devices. To overcome this challenge, in this paper, we advocate the…

Machine Learning · Computer Science 2024-01-25 Zheng Lin , Guangyu Zhu , Yiqin Deng , Xianhao Chen , Yue Gao , Kaibin Huang , Yuguang Fang

We combine concepts from multilevel solvers for partial differential equations (PDEs) with neural network based deep learning and propose a new methodology for the efficient numerical solution of high-dimensional parametric PDEs. An…

Machine Learning · Computer Science 2023-04-05 Cosmas Heiß , Ingo Gühring , Martin Eigel