DAMSL-210 Modeling in Physical Sciences

Syllabus

• Models and their derivation. Dimensional analysis. Examples.
• Population (Logistic, Lotka-Volterra) and disease spreading (SIR) models. Numerical solution of the models.
• Hamiltonian systems (discrete and continuous). Klein-Gordon equation.
• The Schrödinger equation and the Gross-Pitaevskii model.
• Diffusion equation, derivation, properties, diffusion in the presence of constant force.
• Deterministic – Stochastic systems. Modeling of deterministic systems: Molecular dynamics simulations. Dynamics of stochastic systems: Langevin, Brownian dynamics.
• Introduction to Monte Carlo methods. Monte Carlo algorithms of importance sampling.
• Markovian chains. Metropolis – Hastings type algorithms. Examples.
• Molecular simulations and multi-scale modeling of complex systems
• Data-driven models and Machine Learning algorithms for complex systems
• Hybrid physics-based data-driven models
• Advanced Topics: Reaction diffusion systems

Learning Outcomes

After the successful completion of the course the students will be able to:

• Understand the structure of mathematical models for discrete and continuous systems, which appear in Physics, Material Science, Biology and Engineering sciences.
• They will be able to construct  mathematical models for similar problems
• They will be able to use modern simulation methods, algorithms for data analysis and machine learning widely used for the solution of these models
• Be in position to search for solutions to these models with analytical and computational tools
• They will be able to interpreter and  explain the characteristics of the obtained solution with the respect to the origins of the models and its correspondence to the physical problem