cognitive/knowledge_base/mathematics/path_integral_implementations.md

---
type: implementation
id: path_integral_implementations_001
created: 2024-03-15
modified: 2024-03-15
tags: [numerical-methods, algorithms, path-integrals, implementation, computation]
aliases: [path-integral-algorithms, numerical-path-integrals]
complexity: advanced
processing_priority: 1
semantic_relations:
  - type: implements
    links:
      - [[path_integral_free_energy]]
      - [[path_integral_bridge]]
  - type: uses
    links:
      - [[numerical_methods]]
      - [[optimization_theory]]
      - [[differential_geometry]]
  - type: supports
    links:
      - [[active_inference]]
      - [[continuous_time_active_inference]]
---

# Path Integral Implementation Methods

## Numerical Integration Methods

### 1. Symplectic Integration
```python
class SymplecticPathIntegrator:
    """Symplectic integration for path integrals"""
    def __init__(self, order=4):
        self.order = order
        self.coefficients = self._get_coefficients()
        
    def _get_coefficients(self):
        """Get coefficients for symplectic integration"""
        if self.order == 2:
            return {'c': [0.5], 'd': [1.0]}
        elif self.order == 4:
            # Forest-Ruth coefficients
            w = 1.0 / (2 - 2**(1/3))
            return {
                'c': [w/2, (1-w)/2, (1-w)/2, w/2],
                'd': [w, 1-2*w, w]
            }
            
    def integrate(self, hamiltonian, initial_state, dt, steps):
        """Perform symplectic integration"""
        state = initial_state.copy()
        trajectory = [state]
        
        for _ in range(steps):
            # Position updates
            for c in self.coefficients['c']:
                state[::2] += c * dt * hamiltonian.momentum_gradient(state)
                
            # Momentum updates
            for d in self.coefficients['d']:
                state[1::2] -= d * dt * hamiltonian.position_gradient(state)
                
            trajectory.append(state.copy())
            
        return np.array(trajectory)
```

### 2. Stochastic Integration
```python
class StochasticPathIntegrator:
    """Stochastic integration for path integrals"""
    def __init__(self, method='milstein'):
        self.method = method
        self.noise_generator = NoiseGenerator()
        
    def integrate_sde(self, drift, diffusion, initial_state, dt, steps):
        """Integrate stochastic differential equation"""
        state = initial_state.copy()
        trajectory = [state]
        
        for _ in range(steps):
            dW = self.noise_generator.wiener_increment(dt)
            
            if self.method == 'euler':
                # Euler-Maruyama method
                state += drift(state)*dt + diffusion(state)*dW
            elif self.method == 'milstein':
                # Milstein method
                state += (drift(state)*dt + 
                         diffusion(state)*dW +
                         0.5*diffusion(state)*
                         diffusion.derivative(state)*
                         (dW**2 - dt))
                
            trajectory.append(state.copy())
            
        return np.array(trajectory)
```

### 3. Path Sampling
```python
class PathSampler:
    """Path sampling methods for path integrals"""
    def __init__(self, n_samples=1000):
        self.n_samples = n_samples
        self.integrator = StochasticPathIntegrator()
        
    def sample_paths(self, drift, diffusion, initial_state, dt, steps):
        """Sample multiple paths"""
        paths = []
        
        for _ in range(self.n_samples):
            path = self.integrator.integrate_sde(
                drift, diffusion, initial_state, dt, steps)
            paths.append(path)
            
        return np.array(paths)
        
    def importance_sampling(self, action_functional, paths):
        """Perform importance sampling of paths"""
        # Compute action for each path
        actions = np.array([action_functional(path) for path in paths])
        
        # Compute weights
        weights = np.exp(-actions)
        weights /= np.sum(weights)
        
        # Resample paths
        indices = np.random.choice(
            len(paths), size=self.n_samples, p=weights)
        resampled_paths = paths[indices]
        
        return resampled_paths, weights
```

## Optimization Methods

### 1. Path Optimization
```python
class PathOptimizer:
    """Optimization methods for path integrals"""
    def __init__(self, method='natural_gradient'):
        self.method = method
        self.metric_computer = MetricComputer()
        
    def optimize_path(self, initial_path, action_functional, n_steps):
        """Optimize path using gradient descent"""
        path = initial_path.copy()
        
        for _ in range(n_steps):
            # Compute gradient of action
            gradient = self.compute_action_gradient(
                path, action_functional)
                
            if self.method == 'natural_gradient':
                # Compute Fisher metric
                metric = self.metric_computer.fisher_metric(path)
                # Natural gradient update
                path -= np.linalg.solve(metric, gradient)
            else:
                # Standard gradient descent
                path -= gradient
                
        return path
        
    def compute_action_gradient(self, path, action_functional):
        """Compute gradient of action functional"""
        eps = 1e-6
        gradient = np.zeros_like(path)
        
        for i in range(len(path)):
            path_plus = path.copy()
            path_plus[i] += eps
            path_minus = path.copy()
            path_minus[i] -= eps
            
            gradient[i] = (action_functional(path_plus) - 
                         action_functional(path_minus)) / (2*eps)
            
        return gradient
```

### 2. Precision Optimization
```python
class PrecisionOptimizer:
    """Optimization methods for precision parameters"""
    def __init__(self, learning_rate=0.01):
        self.learning_rate = learning_rate
        
    def optimize_precision(self, paths, prediction_errors, n_steps):
        """Optimize precision parameters"""
        precision = np.eye(prediction_errors.shape[1])
        
        for _ in range(n_steps):
            # Compute gradient
            gradient = self.compute_precision_gradient(
                paths, prediction_errors, precision)
                
            # Update precision
            precision += self.learning_rate * gradient
            
            # Ensure positive definiteness
            precision = 0.5 * (precision + precision.T)
            eigvals, eigvecs = np.linalg.eigh(precision)
            eigvals = np.maximum(eigvals, 1e-6)
            precision = eigvecs @ np.diag(eigvals) @ eigvecs.T
            
        return precision
```

## Implementation Utilities

### 1. Numerical Differentiation
```python
class NumericalDifferentiator:
    """Numerical differentiation utilities"""
    def __init__(self, method='central'):
        self.method = method
        
    def derivative(self, func, x, eps=1e-6):
        """Compute numerical derivative"""
        if self.method == 'central':
            return (func(x + eps) - func(x - eps)) / (2*eps)
        elif self.method == 'forward':
            return (func(x + eps) - func(x)) / eps
        elif self.method == 'backward':
            return (func(x) - func(x - eps)) / eps
            
    def jacobian(self, func, x, eps=1e-6):
        """Compute Jacobian matrix"""
        n = len(x)
        jac = np.zeros((n, n))
        
        for i in range(n):
            x_plus = x.copy()
            x_plus[i] += eps
            x_minus = x.copy()
            x_minus[i] -= eps
            
            jac[:, i] = (func(x_plus) - func(x_minus)) / (2*eps)
            
        return jac
```

### 2. Metric Computation
```python
class MetricComputer:
    """Computation of geometric metrics"""
    def __init__(self):
        self.differentiator = NumericalDifferentiator()
        
    def fisher_metric(self, path, log_prob_func):
        """Compute Fisher information metric"""
        # Compute Jacobian of log probability
        jac = self.differentiator.jacobian(log_prob_func, path)
        
        # Fisher metric is expected outer product
        metric = jac.T @ jac
        
        return metric
        
    def path_metric(self, path, lagrangian):
        """Compute path space metric"""
        # Compute second derivatives of Lagrangian
        metric = np.zeros((len(path), len(path)))
        
        for i in range(len(path)):
            for j in range(len(path)):
                metric[i,j] = self.differentiator.derivative(
                    lambda x: self.differentiator.derivative(
                        lambda y: lagrangian(path, i, j),
                        x
                    ),
                    path[i]
                )
                
        return metric
```

### 3. Action Functionals
```python
class ActionFunctionals:
    """Collection of action functionals"""
    def __init__(self):
        self.differentiator = NumericalDifferentiator()
        
    def classical_action(self, path, lagrangian, dt):
        """Compute classical action"""
        action = 0.0
        
        for i in range(len(path)-1):
            velocity = (path[i+1] - path[i]) / dt
            action += lagrangian(path[i], velocity) * dt
            
        return action
        
    def quantum_action(self, path, potential, mass, dt):
        """Compute quantum action"""
        action = 0.0
        
        for i in range(len(path)-1):
            # Kinetic term
            velocity = (path[i+1] - path[i]) / dt
            kinetic = 0.5 * mass * np.sum(velocity**2)
            
            # Potential term
            potential_energy = potential(path[i])
            
            action += (kinetic - potential_energy) * dt
            
        return action
```

## Validation Methods

### 1. Conservation Laws
```python
class ConservationChecker:
    """Check conservation laws"""
    def __init__(self, tolerance=1e-6):
        self.tolerance = tolerance
        
    def check_energy_conservation(self, path, hamiltonian):
        """Check energy conservation"""
        energies = [hamiltonian(state) for state in path]
        energy_std = np.std(energies)
        
        return energy_std < self.tolerance
        
    def check_momentum_conservation(self, path):
        """Check momentum conservation"""
        momenta = path[:, 1::2]  # Extract momentum components
        momentum_std = np.std(np.sum(momenta, axis=1))
        
        return momentum_std < self.tolerance
```

### 2. Error Analysis
```python
class ErrorAnalyzer:
    """Analyze numerical errors"""
    def __init__(self):
        pass
        
    def compute_local_error(self, exact_path, numerical_path):
        """Compute local truncation error"""
        return np.max(np.abs(exact_path - numerical_path))
        
    def compute_global_error(self, exact_path, numerical_path):
        """Compute global accumulation error"""
        return np.sqrt(np.mean((exact_path - numerical_path)**2))
        
    def convergence_rate(self, dt_values, errors):
        """Compute convergence rate"""
        return np.polyfit(np.log(dt_values), np.log(errors), 1)[0]
```

## References
- [[hairer_2006]] - "Geometric Numerical Integration"
- [[leimkuhler_2004]] - "Simulating Hamiltonian Dynamics"
- [[kloeden_1992]] - "Numerical Solution of Stochastic Differential Equations"

## See Also
- [[path_integral_free_energy]]
- [[path_integral_bridge]]
- [[numerical_methods]]
- [[optimization_theory]]
- [[differential_geometry]]