Advancing Partial Differential Equations Solutions: Numerical Methods Meet Machine Learning
A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".
Deadline for manuscript submissions: 27 February 2026 | Viewed by 24
Special Issue Editor
Interests: numerical calculation methods for partial differential equations; discontinuous Galerkin finite element method; fractional-order differential equations; mechanism and data fusion calculation; data assimilation algorithm
Special Issue Information
Dear Colleagues,
This Special Issue explores recent advances in solving partial differential equations (PDEs) and their inverse problems through both traditional numerical methods and emerging machine learning approaches. It highlights developments in classical techniques—finite element, finite difference, and spectral methods—while showcasing innovative data-driven methodologies, particularly physics-informed neural networks and operator learning frameworks. The collection examines the design of computationally efficient algorithms that leverage machine learning to overcome challenges in complex PDE systems. Contributors provide rigorous theoretical analysis, focusing on stability properties, convergence rates, and error estimation for both conventional and novel approaches. By bridging established numerical analysis with cutting-edge machine learning techniques, this Special Issue offers insights into the evolving landscape of computational mathematics. The featured studies demonstrate how this integration enhances our ability to effectively simulate and analyze PDE-governed systems across diverse scientific and engineering applications.
We look forward to your contributions.
Dr. Qinwu Xu
Guest Editor
Manuscript Submission Information
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Keywords
- partial differential equations (PDEs)
- numerical methods
- finite element method (FEM)
- machine learning
- physics-informed neural networks (PINNs)
- inverse problems
- data-driven methods
- computational efficiency
- numerical analysis
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