The Nature of Mathematical Modeling
class: Thursdays 1:00-4:00 E14-493
section: Mondays 5:00-6:00 E14-493
Neil Gershenfeld
MAS 864
2026

When should you use a SVD? SVM? HMM? GMM? CWM? CNN? RNN? PINN? RBF? VAE? GAN? LLM?
When should you apply MD? DEM? MIPS? MPM? PDB? LBM? FEM? PCA? ICA?
When should you employ analytical vs numerical vs data-driven methods?
How do you program 2 cores? 2000 cores? 2000000 cores?
How do you represent: beliefs? expectations? uncertainty?
How do you search: with a model? without a model?

This course will answer these questions, and many more, through a survey of the range of levels of description for mathematical modeling. The focus will be on understanding how these methods relate, and on how they can be implemented efficiently. Projects based on the class have included data analytics, music synthesis, physics simulation, movie special effects, motion control systems, engineering design optimization, and microelectronics signal processing.

The schedule will be:

2/5: Introduction
Mathematical Computing
Linear Algebra and Calculus
2/12: Differential and Difference Equations
2/19: Finite Differences
2/26: Finite Elements
3/5: Discrete Elements
3/12: Random Sysmtes
3/19: Transforms
3/26: Spring Break
4/2: Function Fitting
4/9: Neural Networks
4/16: Search
4/23: State and Density Estimation
4/30: Constrained Optimization
5/7: Machine Learning
5/14: no class
TBA: final exam

Relevant background for each of these areas will be covered. The assignments will include problem sets, programming tasks, and a semester modeling project. The course will use an interactive computing revision of the text The Nature of Mathematical Modeling.

People

Prior