Math Model Selector

Math Model Selector

When to Use

Trigger on phrases like:

• "what math should I use"
• "which mathematical framework"
• "how do I model this"
• "what kind of problem is this"
• "formalize this problem"

Use when user has a problem but doesn't know which mathematical domain applies.

Process

Guide user through decision tree using Polya-style questions:

1. Identify the quantity

Ask: "What quantity or phenomenon are you trying to understand?"

• Physics problem -> conservation laws, differential equations
• Economics -> equilibrium, optimization
• Data patterns -> statistics, regression

2. Characterize change

Ask: "What changes, and how does it change?"

• Discrete steps -> difference equations, recurrences
• Continuous rate -> ODEs
• Rate of rate matters -> 2nd order ODEs
• Spatial variation -> PDEs

3. Check for uncertainty

Ask: "Is there randomness or uncertainty involved?"

• Deterministic -> standard analysis
• Epistemic uncertainty -> Bayesian methods
• Random process -> probability theory, stochastic processes

4. Optimization check

Ask: "Are you optimizing something?"

• Convex objective -> linear/quadratic programming
• Non-convex -> gradient descent, evolutionary methods
• Discrete choices -> combinatorics, integer programming

5. Answer precision

Ask: "What level of answer do you need?"

• Rough estimate -> dimensional analysis
• Qualitative behavior -> phase portraits, stability
• Numerical answer -> simulation
• Exact closed form -> analytical methods

Key Questions to Ask

• What changes? (discrete vs continuous)
• What causes the change? (rate dependencies)
• What's random? (uncertainty type)
• What's being optimized? (objective function)
• How precise? (qualitative vs quantitative)

Output Format

Framework Recommendation:
- Primary: [framework name]
- Why: [one-sentence justification]

Starting Point:
- Key equations: [relevant formulas]
- Initial approach: [first step]

Tools to Use:
- [specific script or computation tool]

Related Skills:
- [domain skill to activate next]

Example Interaction

User: I want to model how a rumor spreads through a population

Claude: Let me help you find the right framework.

Q1: What quantity are we tracking?
> Number of people who heard the rumor over time.

Q2: What causes it to change?
> Spread rate depends on who knows and who doesn't.

Q3: Is there randomness?
> Assume deterministic (average behavior) for now.

Q4: Optimizing anything?
> No, just understanding dynamics.

Q5: What precision?
> Qualitative behavior, maybe predict peak rate.

Framework Recommendation:
- Primary: First-order ODE (logistic growth)
- Why: Rate depends on current state nonlinearly

Starting Point:
- dN/dt = r * N * (1 - N/K)
- N = heard count, K = population, r = spread rate

Tools:
- sympy_compute.py solve (equilibrium)
- Phase line for qualitative behavior

Related Skills:
- math/math-intuition-builder (understand the model)
- odes-pdes/first-order-odes (solve it)

Co-activates

After framework selection, suggest:

• math-intuition-builder - to understand the framework
• Relevant domain skill - for actual computation
• math-mode - for symbolic verification