Deep Euler and Heun Methods:Solving ODEs numerically with deep learning : Djupa Euler- och Heunmetoder: Numeriska lösningar av ordinäradifferentialekvationer med djupinlärning

Sammanfattning

This thesis investigates classical and deep learning-based numerical methods for solving ordinary differential equations (ODEs), with focus on accuracy and numerical stability.We study the Euler and Heun methods for single equations and systems of ODEs, including their error analysis and stability properties. Building on these methods, we introduce the Deep Euler Method (DEM) and Deep Heun Method (DHM), where neural networks are used to learn the local truncation error of the corresponding classical schemes. The proposed deep methods improve numerical accuracy, particularly for larger step sizes,while preserving stability. Numerical simulations and computational experiments were performed on a Continuous Glucose Monitoring (CGM) model involving glucose concentration and elimination dynamics. The computational results demonstrated improved accuracy and stability for larger step sizes in the CGM model. Artificial intelligence tools were used to assist in the computational implementation, Python code development, and numerical simulations carried out in this study.