Mathematical Modeling

Prerequisite(s)

Ordinary Differential Equations

Course description:

This course is designed for upper-level undergraduate or graduate students in mathematics, applied mathematics, physics, and engineering who wish to bridge the gap between classical mathematical physics and modern data science. It is also open to students interested in complex systems and quantitative modeling. The teaching objective is to enable students to master the art of mathematical modeling through a modern lens, integrating first-principle derivations with data-driven perspectives. The course emphasizes the dual nature of modeling—understanding the "microscopic" trajectories (simulation) and the "macroscopic" evolution (analysis).

Topics focus on four core pillars:

• Elementary models (scaling and networks)

• Equation-based models (dynamical systems and conservation laws)

• Optimization theory (variational principles and control)

• Stochastic models (SDEs and generative processes)

This course will employ a variety of teaching methods, including theoretical lectures, computational labs (simulation and visualization), case studies, and a final capstone project connecting theory to modern AI applications.

Learning objectives:

By the end of this course, students will be able to:

• Analyze complex systems using fundamental tools such as dimensional analysis, scaling laws, and network science to extract governing rules from high-dimensional data.

• Construct and evaluate deterministic models based on instantaneous rates of change, utilizing phase plane analysis, bifurcation theory, and switching fluently between Eulerian and Lagrangian perspectives for transport and collective dynamics problems.

• Apply optimization frameworks—from the calculus of variations and Euler-Lagrange equations to Optimal Control and the Hamilton-Jacobi-Bellman equation—to solve physical and decision-making problems, establishing the theoretical link to Reinforcement Learning.

• Model stochastic phenomena using a dual perspective: implementing micro-level simulations (Gillespie algorithm, Langevin dynamics) and deriving macro-level probabilistic evolution equations (Master equation, Fokker-Planck equation).

• Synthesize concepts from differential equations and stochastic processes to understand modern applications, such as the mathematical foundations of diffusion models in generative AI.

Detailed topics covered:

Dynamical Systems & Patterns:

• Phase plane analysis

• Nullclines

• Linear stability

• Jacobian matrices

• Bifurcation theory (saddle-node, Hopf)

• Reaction-diffusion systems

• Turing instability

• Pattern formation

• Dispersion relations

Conservation & Transport:

• Integral conservation laws

• Flux and source terms

• Traffic flow models (LWR)

• Shock waves

• Method of characteristics

• Eulerian vs. Lagrangian specifications

• Material derivative

• Collective dynamics

• Agent-based modeling (Vicsek/Kuramoto)

• Mean-field limits

Optimization & Control:

• Functionals

• Calculus of variations

• Euler-Lagrange equation

• Principle of least action

• Constrained optimization

• Lagrange multipliers

• Minimal surfaces

• Optimal control

• Pontryagin's maximum principle

• Hamiltonians

• Hamilton-Jacobi-Bellman (HJB) equation

• Dynamic programming

• Introduction to Reinforcement Learning

Stochastic Models:

• Monte Carlo methods

• High-dimensional probability intuition

• Discrete-time Markov chains

• Random walks

• MCMC

• Metropolis-Hastings algorithm

• Continuous-time Markov chains

• Poisson processes

• Chemical Master Equation

Stochastic Dynamics & AI:

• Brownian motion

• Stochastic differential equations (Langevin)

• Euler-Maruyama numerical method

• Fokker-Planck equation

• Itô's lemma

• Forward and reverse diffusion processes

• Mathematical foundations of generative diffusion models