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Neural Operators (NOs) provide a powerful framework for computations involving physical laws that can be modelled by (integro-) partial differential equations (PDEs), directly learning maps between infinite-dimensional function spaces that…

Machine Learning · Computer Science 2025-09-18 Gianluca Fabiani , Hannes Vandecasteele , Somdatta Goswami , Constantinos Siettos , Ioannis G. Kevrekidis

Deep neural operators are recognized as an effective tool for learning solution operators of complex partial differential equations (PDEs). As compared to laborious analytical and computational tools, a single neural operator can predict…

Machine Learning · Statistics 2023-02-14 Navaneeth N , Tapas Tripura , Souvik Chakraborty

The solution of partial differential equations (PDEs) plays a central role in numerous applications in science and engineering, particularly those involving multiphase flow in porous media. Complex, nonlinear systems govern these problems…

Accurate urban microclimate analysis with wind velocity and temperature is vital for energy-efficient urban planning, supporting carbon reduction, enhancing public health and comfort, and advancing the low-altitude economy. However,…

Multiphase flow simulation is critical in science and engineering but incurs high computational costs due to complex field discontinuities and the need for high-resolution numerical meshes. While Neural Operators (NOs) offer an efficient…

Fluid Dynamics · Physics 2025-12-04 Zhenzhong Wang , Xin Zhang , Jun Liao , Min Jiang

Operator learning is a variant of machine learning that is designed to approximate maps between function spaces from data. The Fourier Neural Operator (FNO) is one of the main model architectures used for operator learning. The FNO combines…

Numerical Analysis · Mathematics 2025-09-29 Samuel Lanthaler , Andrew M. Stuart , Margaret Trautner

Neural operators are a type of deep architecture that learns to solve (i.e. learns the nonlinear solution operator of) partial differential equations (PDEs). The current state of the art for these models does not provide explicit…

Machine Learning · Computer Science 2022-08-03 Emilia Magnani , Nicholas Krämer , Runa Eschenhagen , Lorenzo Rosasco , Philipp Hennig

We propose an accelerated computational fluid dynamics framework based on a hybrid Fourier Neural Operator-Lattice Boltzmann Method (FNO-LBM) for steady and unsteady weakly compressible flows. FNO-based initialization significantly…

Fluid Dynamics · Physics 2026-05-01 Alexandra Junk , Josef M. Winter , Meike Tütken , Steffen Schmidt , Nikolaus A. Adams

Deep learning surrogate models have shown promise in solving partial differential equations (PDEs). Among them, the Fourier neural operator (FNO) achieves good accuracy, and is significantly faster compared to numerical solvers, on a…

Machine Learning · Computer Science 2024-05-03 Zongyi Li , Daniel Zhengyu Huang , Burigede Liu , Anima Anandkumar

Simulation of urban wind environments is crucial for urban planning, pollution control, and renewable energy utilization. However, the computational requirements of high-fidelity computational fluid dynamics (CFD) methods make them…

Machine Learning · Computer Science 2025-01-13 Cheng Chen , Geng Tian , Shaoxiang Qin , Senwen Yang , Dingyang Geng , Dongxue Zhan , Jinqiu Yang , David Vidal , Liangzhu Leon Wang

Accurately quantifying long-term risk probabilities in diverse stochastic systems is essential for safety-critical control. However, existing sampling-based and partial differential equation (PDE)-based methods often struggle to handle…

Systems and Control · Electrical Eng. & Systems 2025-08-29 Zhuoyuan Wang , Raffaele Romagnoli , Kamyar Azizzadenesheli , Yorie Nakahira

Fourier Neural Operators (FNOs) have been promoted as fast, mesh-invariant surrogates for partial-differential equation solvers, with seismic studies reporting orders-of-magnitude speedup over classical methods. We revisit those claims by…

Geophysics · Physics 2025-08-18 Dimitri Voytan , Litan Li

Neural operators (NOs) provide a new paradigm for efficiently solving partial differential equations (PDEs), but their training depends on costly high-fidelity data from numerical solvers, limiting applications in complex systems. We…

Computational Physics · Physics 2026-05-18 Wen You , Shaoqian Zhou , Xuhui Meng

Predicting the large-scale dynamics of three-dimensional (3D) turbulence is challenging for machine learning approaches. This paper introduces a transformer-based neural operator (TNO) to achieve precise and efficient predictions in the…

Fluid Dynamics · Physics 2024-06-07 Zhijie Li , Tianyuan Liu , Wenhui Peng , Zelong Yuan , Jianchun Wang

Neural operators have proven to be a promising approach for modeling spatiotemporal systems in the physical sciences. However, training these models for large systems can be quite challenging as they incur significant computational and…

Machine Learning · Computer Science 2023-12-12 Michael McCabe , Peter Harrington , Shashank Subramanian , Jed Brown

Combined Sewer Overflow (CSO) is a major problem to be addressed by many cities. Understanding the behavior of sewer system through proper urban hydrological models is an effective method of enhancing sewer system management. Conventional…

Computers and Society · Computer Science 2018-11-16 Duo Zhang , Geir Lindholm , Harsha Ratnaweera

Partial differential equations (PDEs) govern a wide range of physical phenomena, but their numerical solution remains computationally demanding, especially when repeated simulations are required across many parameter settings. Recent…

Machine Learning · Computer Science 2026-05-13 Hamda Hmida , Hsiu-Wen Chang Joly , Youssef Mesri

Neural Operators (NOs) are a powerful deep learning framework designed to learn the solution operator that arise from partial differential equations. This study investigates NOs ability to capture the stiff spatio-temporal dynamics of the…

Machine Learning · Computer Science 2026-03-19 Luca Pellegrini

Discovering hidden physical laws and identifying governing system parameters from sparse observations are central challenges in computational science and engineering. Existing data-driven methods, such as physics-informed neural networks…

Machine Learning · Computer Science 2026-04-16 Dibakar Roy Sarkar , Vijay Kag , Birupaksha Pal , Somdatta Goswami

Reliable digital twins of lithium-ion batteries must achieve high physical fidelity with sub-millisecond speed. In this work, we benchmark three operator-learning surrogates for the Single Particle Model (SPM): Deep Operator Networks…

Machine Learning · Computer Science 2025-08-12 Amir Ali Panahi , Daniel Luder , Billy Wu , Gregory Offer , Dirk Uwe Sauer , Weihan Li
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