The recent publications in the field of mathematical modeling and numerical analysis reveal a strong trend towards the integration of advanced mathematical theories with computational techniques to solve complex problems in finance, biology, and physics. A significant portion of the research focuses on the development of novel numerical methods and algorithms that preserve the intrinsic properties of the systems they model, such as symplecticity, conservation laws, and positivity. These methods are not only theoretically sound but also computationally efficient, making them suitable for high-dimensional problems and long-time simulations.

Another notable direction is the application of machine learning techniques, particularly deep reinforcement learning and score-based models, to enhance the performance of traditional numerical methods. This interdisciplinary approach has led to the discovery of new solutions and improved convergence rates in various contexts, from stochastic differential equations to kinetic equations.

In the realm of financial mathematics, there is a growing interest in exploring the connections between soliton theory and market volatility, offering new perspectives on the modeling of financial instruments. Similarly, in the study of neurodegenerative diseases, innovative mathematical models are being developed to understand the mechanisms of protein spreading and tissue atrophy, with potential implications for therapeutic strategies.

Noteworthy Papers

• Travelling wave solutions of an equation of Harry Dym type arising in the Black-Scholes framework: Introduces a novel non-linear evolution equation derived from the Black-Scholes framework, leveraging soliton theory to explore financial-market volatility.
• Convergence rate of Euler-Maruyama scheme for McKean-Vlasov SDEs with density-dependent drift: Provides a detailed analysis of the convergence rate for the Euler-Maruyama scheme applied to McKean-Vlasov SDEs, contributing to the understanding of weak well-posedness.
• Variational integrators for stochastic Hamiltonian systems on Lie groups: Develops structure-preserving numerical methods for stochastic Hamiltonian systems, with applications to rigid body dynamics and beyond.
• Lévy Score Function and Score-Based Particle Algorithm for Nonlinear Lévy--Fokker--Planck Equations: Proposes a new score function and algorithm for simulating jump-diffusion processes, with applications in biology and finance.
• A coupled mathematical and numerical model for protein spreading and tissue atrophy, applied to Alzheimer's disease: Presents a comprehensive model for studying neurodegenerative diseases, combining mathematical analysis with numerical simulations.
• A positivity-preserving truncated Euler--Maruyama method for stochastic differential equations with positive solutions: Addresses the challenge of preserving positivity in multi-dimensional SDEs, offering a novel numerical method with proven convergence properties.
• Numerical solutions of fixed points in two-dimensional Kuramoto-Sivashinsky equation expedited by reinforcement learning: Demonstrates the effectiveness of combining reinforcement learning with numerical methods to find fixed points in complex dynamical systems.
• A structure-preserving collisional particle method for the Landau kinetic equation: Introduces a computationally efficient method for simulating the Landau equation, preserving essential physical properties.
• Score-Based Metropolis-Hastings Algorithms: Bridges the gap between score-based models and Metropolis-Hastings algorithms, enabling more flexible sampling strategies.
• Solving McKean-Vlasov Equation by deep learning particle method: Leverages deep learning to develop a meshless simulation method for the McKean-Vlasov SDE, overcoming traditional limitations in computational efficiency.