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Mathematics
Special Issues
AI-Driven Computational Methods: Theories, Algorithms and Applications
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AI-Driven Computational Methods: Theories, Algorithms and Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E1: Mathematics and Computer Science".

Deadline for manuscript submissions: 29 May 2026 | Viewed by 1044

Special Issue Editors

Prof. Dr. Fajie Wang

E-Mail Website
Guest Editor
College of Mechanical and Electrical Engineering, Qingdao University, Qingdao 266071, China
Interests: computational mechanics; numerical analysis; boundary element method; meshless method; acoustic propagation; heat and mass transfer
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Ji Lin

E-Mail Website
Guest Editor
College of Mechanics and Engineering Science, Hohai University, Nanjing 211100, China
Interests: solid mechanics; computational mechanics; meshless method; wave propagation
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Yingbin Chai

E-Mail Website
Guest Editor
School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
Interests: computational mechanics; mesh-free methods; advanced finite element methods; vibration and wave propagation in composite structures; acoustic problems; numerical analysis; machine learning

Special Issue Information

Dear Colleagues,

In recent decades, numerical methods have served as the foundation for computational mathematics, providing reliable tools for simulating and solving a wide range of scientific and engineering problems. However, the increasing complexity of real-world systems, which are characterized by large-scale, nonlinear, and multi-physics models, has pushed traditional methods to their limits in terms of accuracy, adaptability, and computational efficiency. Meanwhile, the rapid development of artificial intelligence (AI), supported by the explosive growth of data resources, powerful computing hardware, and breakthroughs in machine learning and neural networks, has opened new possibilities for addressing these challenges. AI not only accelerates simulations, but also enables the creation of hybrid approaches that integrate data-driven intelligence with established numerical frameworks. These developments highlight the necessity of systematically exploring AI-driven computational methods in order to both advance theoretical foundations and expand practical capabilities in scientific computation.

This Special Issue of Mathematics (MPDI), “AI-Driven Computational Methods: Theories, Algorithms and Applications”, aims to collect cutting-edge research at the intersection of AI and numerical computation, showcasing innovations in theory, algorithm development, simulation strategies, and applications across scientific disciplines. We invite original research articles and comprehensive reviews that present novel methodologies or significant applications of AI in numerical and computational methods. Topics of interest include, but are not limited to, neural-network-enhanced numerical methods; machine learning for PDEs and ODEs; data-driven finite element and finite difference methods; surrogate modeling and reduced-order models; meshless and particle-based methods with AI support; AI-assisted optimization and inverse problems; convergence and error analysis of AI-integrated schemes; adaptive algorithms guided by AI or reinforcement learning; high-performance computing for AI-based simulations; hybrid frameworks combining traditional solvers with AI models; and applications in computational mechanics, physics-based simulatons, biology, and engineering.

Prof. Dr. Fajie Wang
Prof. Dr. Ji Lin
Prof. Dr. Yingbin Chai
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • AI-driven computational methods
  • machine learning for numerical analysis
  • neural network algorithms
  • data-driven PDE solvers
  • physics-informed neural networks
  • generative adversarial networks
  • hybrid methods combining traditional numerical approaches and AI
  • surrogate models
  • optimization
  • adaptive algorithms
  • error estimation
  • high-performance computing
  • computational mechanics
  • scientific computing

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Published Papers (2 papers)

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Research

45 pages, 17121 KB  
Article
From Black Box to Transparency: An Explainable Machine Learning (ML) Framework for Ocean Wave Prediction Using SHAP and Feature-Engineering-Derived Variable
by Ahmet Durap
Mathematics 2025, 13(24), 3962; https://doi.org/10.3390/math13243962 - 12 Dec 2025
Viewed by 160
Abstract
Accurate prediction of significant wave height (SWH) is central to coastal ocean dynamics, wave–climate assessment, and operational marine forecasting, yet many high-performing machine-learning (ML) models remain opaque and weakly connected to underlying wave physics. We propose an explainable, feature engineering-guided ML framework for [...] Read more.
Accurate prediction of significant wave height (SWH) is central to coastal ocean dynamics, wave–climate assessment, and operational marine forecasting, yet many high-performing machine-learning (ML) models remain opaque and weakly connected to underlying wave physics. We propose an explainable, feature engineering-guided ML framework for coastal SWH prediction that combines extremal wave statistics, temporal descriptors, and SHAP-based interpretation. Using 30 min buoy observations from a high-energy, wave-dominated coastal site off Australia’s Gold Coast, we benchmarked seven regression models (Linear Regression, Decision Tree, Random Forest, Gradient Boosting, Support Vector Regression, K-Nearest Neighbors, and Neural Networks) across four feature sets: (i) Base (Hmax, Tz, Tp, SST, peak direction), (ii) Base + Temporal (lags, rolling statistics, cyclical hour/month encodings), (iii) Base + a physics-informed Wave Height Ratio, WHR = Hmax/Hs, and (iv) Full (Base + Temporal + WHR). Model skill is evaluated for full-year, 1-month, and 10-day prediction windows. Performance was assessed using R2, RMSE, MAE, and bias metrics, with the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) employed for multi-criteria ranking. Inclusion of WHR systematically improves performance, raising test R2 from a baseline range of ~0.85–0.95 to values exceeding 0.97 and reducing RMSE by up to 86%, with a Random Forest|Base + WHR configuration achieving the top TOPSIS score (1.000). SHAP analysis identifies WHR and lagged SWH as dominant predictors, linking model behavior to extremal sea states and short-term memory in the wave field. The proposed framework demonstrates how embedding simple, physically motivated features and explainable AI tools can transform black-box coastal wave predictors into transparent models suitable for geophysical fluid dynamics, coastal hazard assessment, and wave-energy applications. Full article
(This article belongs to the Special Issue AI-Driven Computational Methods: Theories, Algorithms and Applications)
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19 pages, 3659 KB  
Article
Boundary Knot Neural Networks for the Inverse Cauchy Problem of the Helmholtz Equation
by Renhao Wang, Fajie Wang, Xin Li and Lin Qiu
Mathematics 2025, 13(18), 3029; https://doi.org/10.3390/math13183029 - 19 Sep 2025
Cited by 1 | Viewed by 688
Abstract
The traditional boundary knot method (BKM) has certain advantages in solving Helmholtz equations, but it still faces the difficulty of solving ill-posed problems when dealing with inverse problems. This work proposes a novel deep learning framework, the boundary knot neural networks (BKNNs), for [...] Read more.
The traditional boundary knot method (BKM) has certain advantages in solving Helmholtz equations, but it still faces the difficulty of solving ill-posed problems when dealing with inverse problems. This work proposes a novel deep learning framework, the boundary knot neural networks (BKNNs), for solving inverse Cauchy problems of the Helmholtz equation. The method begins by uniformly distributing collocation points on the physical boundary, then employs a fully connected neural network to approximate the source point coefficient vector in the BKM. The physical quantities on the computational domain can be expressed by the BKM formula, and the loss functions can be constructed via accessible conditions on measurable boundaries. After that, the optimal weights and biases can be obtained by training the fully connected neural network, and thus, the source point coefficient vector can be successfully solved. As a machine learning-based meshless scheme, the BKNN eliminates tedious procedures like meshing and numerical integration while handling inverse Cauchy problems with complex boundaries. More importantly, the method itself is an optimization algorithm that completely avoids the complex processing techniques for ill-conditioned problems in traditional methods. Numerical experiments validate the efficacy of the proposed method, showcasing its superior performance over the traditional BKM for solving the Helmholtz equation’s inverse Cauchy problems. Full article
(This article belongs to the Special Issue AI-Driven Computational Methods: Theories, Algorithms and Applications)
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