Introduction¶

The Nature of Mathematical Modeling (draft)¶

Background¶

• The original Cambridge University Press text
  • was written in response to encountering issues in discipline-centric modeling
  • is a survey with three goals
    • introduce concepts
    • implement examples
    • provide pointers for further study
• The second edition was not converging
  • core concepts were largely unchanged
  • there was an ongoing proliferation of tools and applications
• This version (V2) replaces the static text with dynamic notebooks

Topics¶

• Introduction
• Mathematical Computing
• Linear Algebra and Calculus
• Differential and Difference Equations
• Finite Differences
• Finite Elements
• Discrete Elements
• Random Systems
• Transforms
• Function Fitting
• Neural Networks
• Search
• State and Density Estimation
• Constrained Optimization
• Machine Learning

Applications¶

Projects for and from the class have included:

• Data analytics
• Music synthesis
• Physics simulations
• Movie special effects
• Motion control systems
• Engineering design optimization
• Microelectronics signal processing

Questions¶

Questions that will be answered:

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

Types of Models¶

Spaces that will be explored:

import matplotlib.pyplot as plt
pitch = 0.4
width = 7
height = 3
margin = 1
fig,ax = plt.subplots(figsize=(width,height))
ax.axis([0,width,0,height])
ax.set_axis_off()
def left_text(text,row):
    ax.text(margin,height-row*pitch,text,
        ha='right',va='center',
        fontsize=0.5*pitch*72,color='black')
def right_text(text,row):
    ax.text(width-margin,height-row*pitch,text,
        ha='left',va='center',
        fontsize=0.5*pitch*72,color='black')
def arrow(row):
    ax.annotate('',xy=(1.15*margin,height-row*pitch),xytext=(width-2*margin,height-row*pitch),
        arrowprops=dict(color='black',width=1.5,headwidth=8,headlength=14))
    ax.annotate('',xy=(width-1.15*margin,height-row*pitch),xytext=(2*margin,height-row*pitch),
        arrowprops=dict(color='black',width=1.5,headwidth=8,headlength=14))
def entry(left,right,row):
    left_text(left,row)
    right_text(right,row)
    arrow(row)
entry('specific','general',1)
entry('estimation','derivation',2)
entry('numerical','analytical',3)
entry('stochastic','deterministic',4)
entry('microscopic','macroscopic',5)
entry('discrete','continuous',6)
entry('qualitative','quantitative',7)
plt.show()

Problems¶

• First edition problems statements are equivalent to large language model prompts
• New V2 problems are more open-ended, requiring judgement and creativity

(c) Neil Gershenfeld 1/23/26