From Differential Equations to Data Science: A Survey on Analytical Methods in Contemporary Problems - Alok Kumar Tiwari (Recent Trends in Mathematics, 2025)
The integration of differential equations and data science methods represents a dynamic and evolving approach to solving contemporary challenges across a wide range of disciplines, including engineering, physics, biology, economics, and finance. Differential equations have long served as fundamental tools for modeling continuous systems and processes, offering powerful insights into the behavior of natural and man-made phenomena. For example, they describe how heat diffuses through materials, how populations grow in ecosystems, or how financial markets fluctuate over time. These equations provide a structured mathematical framework that captures cause-and-effect relationships in a precise and interpretable way. Despite their importance, the increasing complexity of modern problems often exceeds the capacity of traditional analytical and numerical methods. Real-world systems rarely exist in idealized conditions. Noise, uncertainty, and high-dimensional interactions introduce complications that are difficult, if not impossible, to fully capture using classical approaches. For instance, climate models based on differential equations must account for enormous amounts of interacting variables, while biomedical processes involve nonlinear dynamics that are often only partially understood. In such contexts, relying solely on traditional differential equation modeling can limit accuracy and predictive power. This is where data science, particularly through machine learning and big data analytics, has emerged as a transformative complement. Data-driven approaches enable researchers to work with vast amounts of empirical data to uncover hidden patterns, optimize parameters in differential equations, or even discover new model structures. Machine learning can approximate solutions where closed-form analytical answers do not exist, while statistical learning methods can quantify uncertainty in predictions. By integrating these techniques, one can build hybrid models that retain the interpretability of differential equations while benefiting from the adaptability and predictive strength of data science. The synergy between these two domains is particularly impactful in areas such as personalized medicine, where patient-specific data can refine mathematical models of disease progression, or in energy systems, where real-time sensor data improves forecasts of demand and supply. Similarly, in finance, data-enhanced differential models can better capture market volatility and systemic risks. Ultimately, the blending of differential equations with data science represents not just a technical advancement but also a paradigm shift. It enables a deeper and more holistic understanding of complex systems, making it possible to tackle challenges that were previously beyond reach.This review explores the evolving intersection between differential equations and data science, emphasizing how the synergy between these fields enables more accurate and efficient solutions to contemporary problems. It provides an overview of the fundamental role differential equations play in mathematical modeling and examines the challenges posed by high-dimensional, non-linear systems that are often difficult to address using classical methods. Data science, with its capacity to handle large datasets and generate predictive models, has shown great promise in overcoming these challenges. The article focuses on several key areas where data science and differential equations converge, such as the use of machine learning to approximate solutions to differential equations, data-driven modeling techniques, and hybrid models that combine the strengths of both approaches. Additionally, we highlight real-world applications of this integration in fields like fluid dynamics, healthcare, finance, and robotics, showcasing how this multidisciplinary approach is reshaping problem-solving strategies. By exploring these advancements, this review not only illustrates the potential of combining differential equations with data science but also sets the stage for future research in the development of new methodologies and frameworks. As these fields continue to evolve, their collaboration will undoubtedly lead to further breakthroughs, addressing increasingly complex and varied challenges in mathematics and beyond.
Keywords: Machine Learning; computational mathematics; data science; differential equations; hybrid modeling; predictive analytics