Research in Mathematics

Computational Partial Differential Equations and Scientific Machine Learning: Theory, Algorithms, and Applications

Partial Differential Equations (PDEs) are fundamental mathematical tools for describing and analysing complex phenomena in science, engineering, and technology. They govern a wide range of processes, including fluid flow, heat and mass transfer, wave propagation, material deformation, biological dynamics, and environmental systems. Over the past decades, significant advances in computational methods have enabled the numerical solution of increasingly complex PDE models, facilitating scientific discovery and engineering innovation.

More recently, the emergence of Scientific Machine Learning has introduced a new paradigm that combines data-driven learning with established physical laws. By integrating machine learning techniques with PDE-based models, researchers can develop computational frameworks that are more efficient, scalable, and capable of handling high-dimensional and data-intensive problems. Approaches such as physics-informed neural networks, neural operators, surrogate modeling, and digital twins are transforming the way complex systems are simulated, analysed, and optimized.

The convergence of computational PDEs and Scientific Machine Learning represents a rapidly evolving research frontier, offering new opportunities for advancing mathematical theory, numerical algorithms, and practical applications. This interdisciplinary field is expected to play a pivotal role in addressing emerging challenges across engineering, environmental science, energy systems, healthcare, and advanced manufacturing.

The integration of Computational Partial Differential Equations (PDEs) and Scientific Machine Learning (SciML) is of growing importance because many contemporary scientific and engineering challenges involve complex, multiscale, and data-intensive systems that are difficult to analyze using traditional approaches alone. PDE-based models provide a rigorous mathematical framework for describing physical phenomena, but their numerical solution can become computationally expensive, particularly for high-dimensional, nonlinear, or multiphysics problems. At the same time, purely data-driven machine learning models often lack physical interpretability, reliability, and generalizability beyond the data on which they are trained.

The convergence of these two fields offers a powerful solution by combining the strengths of mathematical modeling and artificial intelligence. Scientific machine learning enables faster simulations, efficient surrogate models, improved parameter estimation, and enhanced predictive capabilities while preserving underlying physical laws. Such advances are crucial for applications ranging from climate prediction and renewable energy systems to biomedical engineering, advanced materials, and digital twins. Furthermore, the development of trustworthy, interpretable, and computationally efficient methods can accelerate scientific discovery and support informed decision-making. As computational resources and data availability continue to grow, this interdisciplinary area is expected to become a cornerstone of next-generation scientific computing and engineering innovation.

This Article Collection focuses on recent advances in Computational Partial Differential Equations (PDEs) and Scientific Machine Learning (SciML), two rapidly evolving fields that are reshaping modern mathematical modeling and scientific computing. The collection aims to provide a platform for disseminating innovative research that combines rigorous mathematical analysis, advanced numerical methods, and data-driven approaches for solving complex PDE-governed systems arising in science, engineering, and technology.

Topics of interest include, but are not limited to:

• Finite Element and Spectral Methods
• Boundary Element Method
• Non-linear and Stochastic PDEs
• Fractional and nonlocal PDEs
• Inverse Problems
• Uncertainty Quantification
• Reduced-order Modeling
• PDE-constrained Optimization
• High-performance Scientific Computing

Contributions on physics-informed neural networks (PINNs), neural operators, operator learning, surrogate modeling, data assimilation, equation discovery, digital twins, and explainable AI for scientific computing are particularly encouraged. Applications involving solid and fluid mechanics, heat transfer, materials science, renewable energy, environmental systems, biomedical engineering, and multi-physics problems are also welcome.

This Article Collection welcomes original research articles presenting novel theoretical developments, computational algorithms, and practical applications. Comprehensive review articles, and interdisciplinary studies that bridge mathematics, computational science, and machine learning are also encouraged.

Keywords: Computational PDEs, Neural Operators, Digital Twins and Surrogate Modeling, Physics-Informed Machine Learning, Numerical Methods and Scientific Computing

Manuscript Submissions:

Manuscript submission is open until 31st May 2027.

Please carefully review the journal scope and author submission instructions prior to submitting a manuscript as it will be rejected if it does not fall within the scope of the journal.

All manuscripts submitted to this Article Collection will undergo desk assessment and peer-review as part of our standard editorial process. Manuscripts which do not fall within the scope of the journal will be rejected.

We are able to offer a 10% Discount to all authors, and have a limited number of 20% Discount codes only available for early submissions. It should be noted that discount codes must be entered in at the point of submission as they cannot be applied retroactively, nor can these be combined as only the higher valued discount would be applicable.

To submit your papers to this Article Collection, please:

1. Check "Yes" for the question, "Are you submitting your paper for a specific special issue or article collection?"
2. Select the relevant Article Collection from the drop-down menu under the question, "Computational Partial Differential Equations and Scientific Machine Learning: Theory, Algorithms, and Applications"
3. Enter the Discount Code on the Final Page (Services) in the “Apply Discount Code” box on the right-hand side of the screen, otherwise this will not be applied to your submission, and you could be then subject to the journal’s Full APC Fee

Please contact Christopher Montgomery, Commissioning Editor regarding details on obtaining your discount codes, and with any other queries for this Article Collection.

Article Collection Guest Advisors

Prof. Manoj Sahni is a dedicated and experienced Mathematics teacher and researcher with more than two decades of experience and currently serving as a Professor in the Department of Mathematics, School of Technology, Pandit Deendayal Energy University, Gandhinagar, Gujarat, India. He has published more than 120 research papers in peer-reviewed Journals (SCI/ SCIE/ Scopus) and Conference Proceedings (Scopus Indexed) with reputed publishers like Springer, Elsevier, Taylor and Francis, Wiley, and many more. He has written ten edited books which were published by publishers like Springer, Taylor and Francis, NOVA and River and four are accepted by reputed publishers and has also served as Guest Editor and Editorial Board Member for Special Issues published by reputed international publishers and scholarly journals.

Dr. Puneet Rana is an Assistant Professor of Mathematics at Wenzhou-Kean University. His research interests include nanotechnology, fluid mechanics, heat transfer, nanofluids, soft computing, numerical methods, and thermal stability analysis. He has published more than 90 research papers in reputed international journals and conference proceedings and has received significant citations for his scholarly contributions. In addition to his research activities, he serves as a reviewer for leading SCI-indexed journals. His work contributes significantly to the advancement of applied mathematics and its interdisciplinary applications in engineering and technology.

Dr. Chandra Shekhar Nishad is an accomplished academician, researcher, and educator in the fields of applied mathematics, scientific computing, and fluid mechanics. His research interests primarily focus on numerical simulation of fluid flow problems, computational mathematics, boundary element methods, and mathematical modelling of complex physical systems. Dr. Nishad has made significant contributions to the advancement of computational techniques for solving engineering and scientific problems. His work integrates mathematical theory with modern computational approaches, addressing challenges in fluid dynamics, numerical analysis, and scientific computing.