Generalized/Extended Finite Element Methods, Meshless Methods and Related Developments in Machine Learning

Dear Colleagues,

Generalized/extended finite element methods (GFEMs/XFEMs) and meshless methods (MMs) are utilized to overcome the difficulty of mesh generation using conventional numerical methods, such as FEMs and finite volume methods. In these methods, regular meshes or particles that are independent of the non-smooth properties of fundamental solutions are adopted, and the complexity of the resulting mesh generation is reduced significantly. In recent decades, GFEMs/XFEMs and MMs have enabled extensive developments in engineering computations. Recently, machine learning (ML) methods, e.g., deep neural networks (NNs) and physics-informed NNs (PINNs), have gained significant attention in the field of scientific computing. In general, ML methods construct loss functions based on sampling points so that they can also be viewed as instances of MMs. The objective of this Special Issue is twofold. The first aim is to collate the progress made in using GFEMs/XFEMs, MMs, ML, NNs, and PINNs in engineering computations and numerical solutions to partial differential equations (PDEs). The second aim is to bridge the connection between GFEMs/XFEMs/MMs and ML/NNs to foster potential research directions. All topics related to GFEMs/XFEMs/MMs/ML/NNs are welcome, including but not limited to the following: high-precision algorithms, engineering applications, theoretical and mathematical analysis, software developments, high-performance computations, stability and robustness, construction of enrichments, nonlinearities, singularities, dynamic analysis, large deformations, multi-physics, evolving PDEs, high-dimensional PDEs, optimization techniques, adaptive algorithms, operator learning, and unfitted mesh methods.

Prof. Dr. Qinghui Zhang
Guest Editor

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• generalized finite element methods
• extended finite element methods
• meshless methods
• unfitted mesh methods
• machine learning
• deep learning
• neural networks
• extreme learning machine
• randomized neural networks
• physics-informed neural networks
• optimization
• high dimension
• high-performance computations
• theories