# Mathematical Foundations The mathematical foundations of cognitive phenomena integrate principles from [[variational_methods|variational methods]], [[information_theory|information theory]], and [[dynamical_systems|dynamical systems]] to formalize how cognitive systems perceive, learn, and act. This framework unifies these processes under the [[free_energy_principle|free energy principle]] through hierarchical prediction error minimization. ## Core Framework ### Free Energy Principle 1. **Variational Free Energy** ([[variational_inference|VI formulation]]) ```math F = ∫ q(θ)[ln q(θ) - ln p(o,θ)]dθ = KL[q(θ)||p(θ|o)] - ln p(o) ``` where: - F is free energy - q(θ) is variational density - p(o,θ) is generative model - KL is Kullback-Leibler divergence 1. **Expected Free Energy** ([[path_integral_free_energy|path integral form]]) ```math G(π) = E_{q(o,s|π)}[ln q(s|π) - ln p(o,s|π)] ``` where: - G is expected free energy - π is policy - s is states - p(o,s|π) is predictive model ### Information Theory 1. **Mutual Information** ([[information_theory|IT principles]]) ```math I(X;Y) = ∑P(x,y)log(P(x,y)/P(x)P(y)) ``` where: - I is mutual information - P(x,y) is joint distribution - P(x), P(y) are marginals 1. **Entropy** ([[information_theory|Shannon entropy]]) ```math H(P) = -∑P(x)log P(x) ``` where: - H is entropy - P(x) is probability distribution ## Advanced Mathematical Structures ### Differential Geometry 1. **Riemannian Manifolds** ```math ds² = g_{ij}dx^idx^j ``` where: - g_{ij} is metric tensor - dx^i are coordinate differentials 1. **Parallel Transport** ```math ∇_X Y = ∂_X Y^i + Γ^i_{jk}X^jY^k ``` where: - ∇_X is covariant derivative - Γ^i_{jk} are Christoffel symbols ### Category Theory 1. **Functorial Relationships** ```math F: C → D ``` where: - F is functor - C, D are categories 1. **Natural Transformations** ```math η: F ⇒ G ``` where: - η is natural transformation - F, G are functors ## Dynamical Systems ### State Space Dynamics 1. **Continuous Dynamics** ([[variational_calculus|calculus of variations]]) ```math dx/dt = f(x,u,θ) + w = -∂F/∂x + D∇²x + η(t) ``` where: - x is state vector - u is control input - F is free energy - D is diffusion tensor 1. **Discrete Updates** ([[active_inference_pomdp|POMDP formulation]]) ```math x_{t+1} = g(x_t,u_t,θ) + w_t ``` where: - x_t is state at time t - u_t is control at time t - g is transition function ### Stochastic Processes 1. **Fokker-Planck Equation** ```math ∂p/∂t = -∇·(fp) + (1/2)∇²(Dp) ``` where: - p is probability density - f is drift vector - D is diffusion matrix 1. **Langevin Dynamics** ```math dx = f(x)dt + σdW ``` where: - f(x) is drift term - σ is noise amplitude - dW is Wiener process ## Advanced Control Theory ### Optimal Control 1. **Hamilton-Jacobi-Bellman Equation** ```math -∂V/∂t = min_u[L(x,u) + (∂V/∂x)·f(x,u)] ``` where: - V is value function - L is cost function - f is dynamics 1. **Pontryagin's Maximum Principle** ```math H(x,p,u) = L(x,u) + p·f(x,u) ``` where: - H is Hamiltonian - p is costate - f is dynamics ### Robust Control 1. **H∞ Control** ```math ||T_{zw}||_∞ ≤ γ ``` where: - T_{zw} is transfer matrix - γ is performance bound 1. **Lyapunov Stability** ```math dV/dt ≤ -αV ``` where: - V is Lyapunov function - α is decay rate ## Advanced Probabilistic Methods ### Information Geometry 1. **Fisher Information Metric** ```math g_{ij}(θ) = E[-∂²ln p(x|θ)/∂θ_i∂θ_j] ``` where: - g_{ij} is metric tensor - p(x|θ) is likelihood 1. **Natural Gradient Flow** ```math dθ/dt = -g^{ij}∂F/∂θ_j ``` where: - g^{ij} is inverse metric - F is objective function ### Variational Methods 1. **Wasserstein Distance** ```math W_p(μ,ν) = (inf_γ ∫||x-y||^p dγ(x,y))^{1/p} ``` where: - μ,ν are distributions - γ is transport plan 1. **Normalizing Flows** ```math p_K(x) = p_0(f^{-1}_K ∘...∘ f^{-1}_1(x))|det ∏_{k=1}^K ∂f^{-1}_k/∂x| ``` where: - p_K is transformed density - f_k are invertible maps ## Implementation Framework ### Numerical Methods 1. **Gradient Descent** ([[variational_methods|optimization]]) ```math θ_{t+1} = θ_t - α∇F(θ_t) ``` where: - θ_t is parameter at step t - α is learning rate - ∇F is gradient 1. **Message Passing** ([[variational_inference|belief propagation]]) ```math μ_{t+1} = μ_t + κ∂F/∂μ ``` where: - μ_t is belief at step t - κ is update rate - ∂F/∂μ is belief gradient ### Advanced Optimization 1. **Natural Policy Gradient** ```math θ_{t+1} = θ_t - αF^{-1}∇J(θ_t) ``` where: - F is Fisher information - J is objective - α is step size 1. **Trust Region Methods** ```math max_θ L(θ) s.t. KL[π_θ||π_{θ_old}] ≤ δ ``` where: - L is surrogate objective - KL is trust region constraint - δ is step size ## Applications ### Cognitive Architectures 1. **Hierarchical Processing** ```math F_l = E_q[ln q(s_l) - ln p(s_l|s_{l+1}) - ln p(s_{l-1}|s_l)] ``` where: - F_l is level-specific free energy - s_l is state at level l 1. **Predictive Coding** ```math ε_l = μ_l - g(μ_{l+1}) ``` where: - ε_l is prediction error - μ_l is expectation - g is generative mapping ### Learning Systems 1. **Meta-Learning** ```math θ* = argmin_θ E_τ[L(τ; θ)] ``` where: - θ are meta-parameters - τ are tasks - L is task loss 1. **Active Learning** ```math x* = argmax_x H[y|D,x] ``` where: - x* is query point - H is entropy - D is dataset ## Advanced Topics ### Quantum Information 1. **Von Neumann Entropy** ```math S(ρ) = -Tr(ρ ln ρ) ``` where: - ρ is density matrix - Tr is trace 1. **Quantum Channels** ```math Φ(ρ) = ∑_k E_k ρ E_k^† ``` where: - Φ is channel - E_k are Kraus operators ### Topological Data Analysis 1. **Persistent Homology** ```math β_k(ε) = dim H_k(X_ε) ``` where: - β_k is Betti number - H_k is homology group - X_ε is filtration 1. **Mapper Algorithm** ```math M(X,f,U,C) = N(f^{-1}(U),C) ``` where: - X is dataset - f is filter function - U is cover - C is clustering ## Future Directions ### Emerging Frameworks 1. **Geometric Deep Learning** - Group equivariance - Manifold learning - Graph neural networks 1. **Causal Learning** - Structural equations - Intervention calculus - Counterfactual reasoning ### Open Problems 1. **Theoretical Challenges** - Scale separation - Non-equilibrium dynamics - Information bottlenecks 1. **Practical Challenges** - Computational efficiency - Model interpretability - Robustness guarantees ## Related Concepts - [[variational_methods]] - [[variational_calculus]] - [[variational_inference]] - [[active_inference]] - [[free_energy_principle]] ## References - [[jordan_1999]] - "Introduction to Variational Methods" - [[friston_2010]] - "The Free-Energy Principle" - [[amari_2000]] - "Information Geometry" - [[parr_friston_2019]] - "Generalised Free Energy"