Approximation Assistant
Model: Interconnected Investment Model
Objective: Equation simplification and asymptotic reduction
Detected agents: 12
Detected Minimization Equations
min Eᵢ = ∑ⱼ Jᵢⱼ (xᵢ - xⱼ)² + λxᵢ² - hᵢxᵢ
min F(ρ) = ∫ V(x)ρ(x)dx + α∫∫ W(x,y)ρ(x)ρ(y)dxdy
Extracted Parameters
Jᵢⱼ : interaction matrix
λ : regularization coefficient
hᵢ : external forcing
α : interaction strength
V(x) : local potential
W(x,y) : pair interaction kernel
Approximation Checklist
- ✔All minimization equations identified
- ✔Parameters extracted
- Constant terms identified
- Negligible terms identified
- Functional forms proposed
- Reduced approximation validated
Approximation AI
I analyzed the minimization equations. Several terms may admit simplifying assumptions or asymptotic approximations.
Proposed Approximations
Mean-field approximation
Replace local interactions by averaged interaction terms.
Small fluctuation regime
Assume second-order fluctuations remain negligible.
Reduced Approximate Model
Assume λ = constant
Neglect O(δ²) fluctuation terms
Linearized interaction:
F(x) ≈ ax + b
∂ρ/∂t = -∇·(ρ(ax+b)) + DΔρ