Applications of Ordinary and Partial Differential Equations in Machine Learning and Data Science

Differential equations have emerged as fundamental mathematical tools in machine learning and data science, providing both theoretical foundations and practical methodologies for solving complex problems. This comprehensive review explores the applications of ordinary differential equations (ODEs) and partial differential equations (PDEs) across machine learning domains, including neural network optimization, generative modeling, and physics-informed learning. We discuss continuous-time perspectives on gradient descent, Neural Ordinary Differential Equations as continuous-depth models, diffusion-based generative modeling through stochastic differential equations, Physics-Informed Neural Networks for scientific computing, and flow-based models. This paper demonstrates how differential equations bridge classical mathematics and modern machine learning, providing powerful theoretical insights and practical improvements in model development and training.

• Ordinary differential equations
• Partial differential equations
• Neural networks
• Optimization
• Physics-informed learning
• Generative models
• Stochastic differential equations

• Amit Gupta, Research Scholar, Sharda University, Greater Noida

A. Gupta, “Applications of ordinary and partial differential equations in machine learning and data science,” IPEM Journal of Computer Application & Research, vol. 10, pp. 114–121, Dec. 2025doi: