Numerical Functional Analysis and Optimization

In recent years, the interaction between data-driven and knowledge-driven methods is enjoying strong success within scientific communities, and especially in computational mathematics, in cases when the knowledge of the modelling is not fully available, or whenever the standard methods are computationally unfeasible. The use of machine learning has revolutionized several fields, including the numerical analysis of partial differential equations (PDEs), inverse problems, and compressed sensing. An essential contribution has been given by applied harmonic analysis, in the development of new ideas and methods in signal processing and approximation theory, providing an important tool to obtain a theoretical framework for machine learning theories.

The main goal of this Special Issue is to collect research contributes on various themes where analysis and machine learning play a crucial role, and where data- and physics-driven approaches are utilized, including PDEs, inverse problems, imaging, optimal control, and applied harmonic analysis. Possible topics are the theoretical and numerical analysis of data-driven regularization techniques, network architectures, neural operators for PDEs, deep generative models, and real-world applications of machine learning to the biomedical, engineering, and physical sciences.

Recommended topics include:

• Machine Learning for PDEs
• Data-driven regularization techniques
• Applied Harmonic Analysis
• Machine learning in infinite-dimensional spaces
• Analysis of Neural Networks
• Approximation theory
• Reinforcement learning
• Compressed sensing

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