Mathematical Foundations and Programming Languages: Advancing Machine Learning and Neural Networks

Dear Colleagues,

This Special Issue addresses the growing need for rigorous mathematical and programming language foundations in the design, analysis, and deployment of modern machine learning (ML) systems and neural networks. With the rise in large language models (LLMs), autonomous agents, and embodied intelligence systems, it is essential to revisit the theoretical and algorithmic underpinnings that ensure these models are efficient, interpretable, and aligned with real-world constraints.

We aim to attract contributions that enhance the mathematical expressiveness and computational soundness of ML systems through innovations in optimization theory, non-convex approximation, probabilistic reasoning, and domain-specific programming paradigms.

Topics of interest include (but are not limited to) the following:

• Mathematical Foundations:
• Convergence analysis and generalization bounds of deep models;
• Non-convex approximation theory in high-dimensional optimization;
• Information-theoretic bounds and statistical complexity;
• Manifold learning, geometric deep learning, and topology-aware models.

• Programming Languages for ML:
• Domain-specific languages (DSLs) for differentiable programming;
• Type systems and logic frameworks for verifiable ML pipelines;
• Program synthesis and symbolic reasoning integration;
• Formal verification and safety guarantees in ML codebases.

• Optimization Methods:
• Scalable methods for large-scale non-convex optimization;
• Sparse regularization, bi-level optimization, and meta-learning;
• Gradient compression, quantization, and communication-efficient learning;
• Optimization in hybrid symbolic–neural systems and embodied settings.

• Advanced Neural Architectures and Learning Paradigms:
• Large language models (LLMs)—training dynamics, model scaling laws, fine-tuning theory;
• Intelligent agents—decision-making under uncertainty, multi-agent reinforcement learning;
• Embodied Intelligence—sensorimotor learning, physics-informed neural networks (PINNs), grounded reasoning;
• Graph neural networks, memory-augmented networks, and neural-symbolic systems.

• Applications and Theoretical Innovations:
• Interpretable AI, causality-aware learning, and algorithmic fairness;
• Foundational advances in deep generative modeling and energy-based learning;
• Formal abstractions for aligning LLM behavior with logical constraints;
• Theoretical models of agent cognition and embodied task execution.

Dr. Xiaofeng Cao
Guest Editor

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 100 words) can be sent to the Editorial Office for announcement on this website.

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.

• optimization
• non-convex approximation
• machine learning
• information-theoretic bound
• geometric deep learning
• large language models