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Solving parametric partial differential equations (PDEs) and associated PDE-based, inverse problems is a central task in engineering and physics, yet existing neural operator methods struggle with high-dimensional, discontinuous inputs and…

Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from…

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,…

Fourier neural operators (FNOs) can learn highly nonlinear mappings between function spaces, and have recently become a popular tool for learning responses of complex physical systems. However, to achieve good accuracy and efficiency, FNOs…

Scientific machine learning has seen significant progress with the emergence of operator learning. However, existing methods encounter difficulties when applied to problems on unstructured grids and irregular domains. Spatial graph neural…

This study used a multigrid-based convolutional neural network architecture known as MgNet in operator learning to solve numerical partial differential equations (PDEs). Given the property of smoothing iterations in multigrid methods where…

Neural operators have emerged as powerful tools for learning mappings between function spaces, enabling efficient solutions to partial differential equations across varying inputs and domains. Despite the success, existing methods often…

For partial differential equations on domains of arbitrary shapes, existing works of neural operators attempt to learn a mapping from geometries to solutions. It often requires a large dataset of geometry-solution pairs in order to obtain a…

A computed approximation of the solution operator to a system of partial differential equations (PDEs) is needed in various areas of science and engineering. Neural operators have been shown to be quite effective at predicting these…

Learning dynamics governed by differential equations is crucial for predicting and controlling the systems in science and engineering. Neural Ordinary Differential Equation (NODE), a deep learning model integrated with differential…

The neural operator (NO) framework has emerged as a powerful tool for solving partial differential equations (PDEs). Recent NOs are dominated by the Transformer architecture, which offers NOs the capability to capture long-range…

Neural operators (NO) are discretization invariant deep learning methods with functional output and can approximate any continuous operator. NO have demonstrated the superiority of solving partial differential equations (PDEs) over other…

Transformer-based models have recently shown success in representation learning on graph-structured data beyond natural language processing and computer vision. However, the success is limited to small-scale graphs due to the drawbacks of…

Solving partial differential equations (PDEs) efficiently and accurately remains a cornerstone challenge in science and engineering, especially for problems involving complex geometries and limited labeled data. We introduce a Physics- and…

Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of…

Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work…

Partial differential equations (PDEs) govern diverse physical phenomena, yet high-fidelity numerical solutions are computationally expensive and Machine Learning approaches lack generalization. While Scientific Foundation Models (SFMs) aim…

Neural operators have emerged as powerful data-driven solvers for PDEs, offering substantial acceleration over classical numerical methods. However, existing transformer-based operators still face critical challenges when modeling PDEs on…

Physics-informed neural operators offer a powerful framework for learning solution operators of partial differential equations (PDEs) by combining data and physics losses. However, these physics losses rely on derivatives. Computing these…

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…