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We present a novel deep learning-based algorithm to accelerate - through the use of Artificial Neural Networks (ANNs) - the convergence of Algebraic Multigrid (AMG) methods for the iterative solution of the linear systems of equations…

Numerical Analysis · Mathematics 2025-06-18 Paola F. Antonietti , Matteo Caldana , Luca Dede'

We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral…

Machine Learning · Computer Science 2024-01-22 Yiheng Du , Nithin Chalapathi , Aditi Krishnapriyan

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

One approach to deal with the statistical inefficiency of neural networks is to rely on auxiliary losses that help to build useful representations. However, it is not always trivial to know if an auxiliary task will be helpful for the main…

This paper presents machine learning techniques and deep reinforcement learningbased algorithms for the efficient resolution of nonlinear partial differential equations and dynamic optimization problems arising in investment decisions and…

Optimization and Control · Mathematics 2021-04-19 Maximilien Germain , Huyên Pham , Xavier Warin

Solving time-dependent partial differential equations (PDEs) that exhibit sharp gradients or local singularities is computationally demanding, as traditional physics-informed neural networks (PINNs) often suffer from inefficient point…

Numerical Analysis · Mathematics 2026-01-27 Beining Xu , Haijun Yu , Jiayu Zhai , Kejun Tang , Xiaoliang Wan

We are interested in the approximation of partial differential equations with a data-driven approach based on the reduced basis method and machine learning. We suppose that the phenomenon of interest can be modeled by a parametrized partial…

Numerical Analysis · Computer Science 2020-06-24 Niccolò Dal Santo , Simone Deparis , Luca Pegolotti

Simulations of complex physical systems are typically realized by discretizing partial differential equations (PDEs) on unstructured meshes. While neural networks have recently been explored for surrogate and reduced order modeling of PDE…

Machine Learning · Computer Science 2021-10-27 Jiayang Xu , Aniruddhe Pradhan , Karthik Duraisamy

A growing body of literature has been leveraging techniques of machine learning (ML) to build novel approaches to approximating the solutions to partial differential equations. Noticeably absent from the literature is a systematic…

Numerical Analysis · Mathematics 2026-05-19 Jonah A. Reeger

Modeling dynamics in the form of partial differential equations (PDEs) is an effectual way to understand real-world physics processes. For complex physics systems, analytical solutions are not available and numerical solutions are…

Numerical Analysis · Mathematics 2024-01-19 Zijiang Yang , Zhongwei Qiu , Dongmei Fu

Deep neural networks have recently drawn considerable attention to build and evaluate artificial learning models for perceptual tasks. Here, we present a study on the performance of the deep learning models to deal with global optimization…

Neural and Evolutionary Computing · Computer Science 2020-12-18 Hojjat Rakhshani , Lhassane Idoumghar , Soheila Ghambari , Julien Lepagnot , Mathieu Brévilliers

We propose a new method for training a supervised source separation system that aims to learn the interdependent relationships between all combinations of sources in a mixture. Rather than independently estimating each source from a mix, we…

Sound · Computer Science 2022-03-30 Ethan Manilow , Curtis Hawthorne , Cheng-Zhi Anna Huang , Bryan Pardo , Jesse Engel

This short, self-contained article seeks to introduce and survey continuous-time deep learning approaches that are based on neural ordinary differential equations (neural ODEs). It primarily targets readers familiar with ordinary and…

Machine Learning · Computer Science 2024-01-09 Lars Ruthotto

Deep neural networks conventionally employ end-to-end backpropagation for their training process, which lacks biological credibility and triggers a locking dilemma during network parameter updates, leading to significant GPU memory use.…

Computer Vision and Pattern Recognition · Computer Science 2024-08-13 Junhao Su , Changpeng Cai , Feiyu Zhu , Chenghao He , Xiaojie Xu , Dongzhi Guan , Chenyang Si

Partial differential equations (PDEs) are indispensable for modeling many physical phenomena and also commonly used for solving image processing tasks. In the latter area, PDE-based approaches interpret image data as discretizations of…

Machine Learning · Computer Science 2018-12-12 Lars Ruthotto , Eldad Haber

We review recent machine-learning (ML) approaches for point defects in non-metallic materials, with an emphasis on defect formation energies. Existing studies largely fall into two categories: direct ML models that predict defect energetics…

Materials Science · Physics 2026-05-19 Yu Kumagai , Shin Kiyohara

Motivated by distributed machine learning settings such as Federated Learning, we consider the problem of fitting a statistical model across a distributed collection of heterogeneous data sets whose similarity structure is encoded by a…

Statistics Theory · Mathematics 2021-11-30 Dominic Richards , Sahand N. Negahban , Patrick Rebeschini

Many engineering processes can be accurately modelled using partial differential equations (PDEs), but high dimensionality and non-convexity of the resulting systems pose limitations on their efficient optimisation. In this work, a model…

Optimization and Control · Mathematics 2024-10-17 Min Tao , Panagiotis Petsagkourakis , Jie Li , Constantinos Theodoropoulos

Finite element discretizations of problems in computational physics often rely on adaptive mesh refinement (AMR) to preferentially resolve regions containing important features during simulation. However, these spatial refinement strategies…

Computational Engineering, Finance, and Science · Computer Science 2022-09-27 Corbin Foucart , Aaron Charous , Pierre F. J. Lermusiaux

This paper describes the use of neural networks to enhance simulations for subsequent training of anomaly-detection systems. Simulations can provide edge conditions for anomaly detection which may be sparse or non-existent in real-world…

Neural and Evolutionary Computing · Computer Science 2020-11-06 Philip Feldman