cognitive/knowledge_base/mathematics/path_integral_implementation.md

type	id	created	modified	tags	aliases
implementation	path_integral_implementation_001	2024-02-05	2024-02-05	implementation numerical-methods path-integrals active-inference	path-implementation numerical-path-methods

Implementation Guide for Path Integral Methods

Numerical Foundations

Discretization Framework

class PathDiscretizer:
    """Framework for path discretization schemes."""
    
    def __init__(self, dt: float, scheme: str = 'symplectic'):
        self.dt = dt
        self.scheme = self._initialize_scheme(scheme)
        
    def discretize_action(self, 
                         lagrangian: Callable,
                         path: np.ndarray) -> float:
        """Discretize continuous action functional."""
        discrete_points = self.scheme(path, self.dt)
        return self.compute_discrete_action(lagrangian, discrete_points)

Links to:

• numerical_discretization
• finite_elements
• discrete_mechanics

Stability Analysis

def analyze_stability(scheme: Callable,
                     test_system: Callable,
                     params: Dict) -> Dict[str, float]:
    """Analyze numerical stability of discretization."""
    return {
        'local_error': compute_local_error(scheme, test_system),
        'global_error': compute_global_error(scheme, test_system),
        'energy_drift': compute_energy_conservation(scheme, test_system)
    }

Links to:

• numerical_stability
• error_analysis
• conservation_laws

Path Sampling Methods

MCMC Implementation

class PathMCMC:
    """Markov Chain Monte Carlo for path sampling."""
    
    def __init__(self, action: Callable, temperature: float):
        self.action = action
        self.beta = 1.0 / temperature
        
    def hamiltonian_mc(self,
                      initial_path: np.ndarray,
                      num_steps: int) -> List[np.ndarray]:
        """Hamiltonian Monte Carlo sampling."""
        paths = []
        current = initial_path
        momentum = self.sample_momentum(current.shape)
        
        for _ in range(num_steps):
            # Leapfrog integration
            next_path, next_momentum = self.leapfrog_step(
                current, momentum)
            
            # Metropolis acceptance
            if self.accept_state(current, next_path,
                               momentum, next_momentum):
                current = next_path
                momentum = next_momentum
            
            paths.append(current.copy())
        
        return paths

Links to:

• hamiltonian_monte_carlo
• leapfrog_integration
• metropolis_hastings

Variational Methods

class VariationalPathSampler:
    """Variational inference for path distributions."""
    
    def optimize_variational_parameters(self,
                                     target_density: Callable,
                                     variational_family: str,
                                     **params) -> Dict:
        """Optimize variational approximation."""
        elbo_history = []
        current_params = self.initialize_parameters(variational_family)
        
        while not self.converged(elbo_history):
            # Compute ELBO gradient
            grad = self.compute_elbo_gradient(
                current_params, target_density)
            
            # Update parameters
            current_params = self.update_parameters(
                current_params, grad, **params)
            
            # Track progress
            elbo = self.compute_elbo(
                current_params, target_density)
            elbo_history.append(elbo)
        
        return current_params

Links to:

• variational_inference
• elbo_optimization
• natural_gradients

Active Inference Specifics

Free Energy Computation

class PathFreeEnergy:
    """Compute free energy for path ensembles."""
    
    def __init__(self, model_density: Callable):
        self.model = model_density
        
    def compute_path_free_energy(self,
                               paths: np.ndarray,
                               variational_density: Callable) -> float:
        """Compute free energy for path ensemble."""
        # Energy term
        energy = self.compute_path_energy(paths)
        
        # Entropy term
        entropy = self.compute_path_entropy(
            paths, variational_density)
        
        return energy - entropy

Links to:

• free_energy_principle
• path_entropy
• energy_functionals

Policy Optimization

class PathBasedPolicyOptimizer:
    """Optimize policies using path integral methods."""
    
    def optimize_policy(self,
                       initial_state: np.ndarray,
                       horizon: int,
                       num_samples: int) -> Callable:
        """Optimize policy using path sampling."""
        # Generate path ensemble
        paths = self.sample_paths(
            initial_state, horizon, num_samples)
        
        # Compute path weights
        weights = self.compute_path_weights(paths)
        
        # Update policy
        return self.update_policy_parameters(paths, weights)

Links to:

• policy_gradients
• path_integral_control
• importance_sampling

Advanced Implementation Topics

Parallel Computing

class ParallelPathSampler:
    """Parallel implementation of path sampling."""
    
    def parallel_sample_paths(self,
                            num_paths: int,
                            num_workers: int) -> np.ndarray:
        """Generate paths in parallel."""
        with ProcessPoolExecutor(max_workers=num_workers) as executor:
            paths = list(executor.map(
                self.generate_single_path,
                range(num_paths)))
        return np.array(paths)

Links to:

• parallel_processing
• distributed_computing
• gpu_acceleration

Adaptive Methods

class AdaptivePathSampling:
    """Adaptive sampling methods for paths."""
    
    def adapt_step_size(self,
                       acceptance_rate: float,
                       current_step: float) -> float:
        """Adapt integration step size."""
        target_rate = 0.65  # Optimal acceptance rate
        log_step = np.log(current_step)
        return np.exp(log_step + 
                     self.learning_rate * 
                     (acceptance_rate - target_rate))

Links to:

• adaptive_mcmc
• dual_averaging
• step_size_adaptation

Validation and Testing

Unit Tests

class PathIntegralTests:
    """Test suite for path integral implementations."""
    
    def test_energy_conservation(self):
        """Test energy conservation in sampling."""
        sampler = PathSampler(self.test_system)
        paths = sampler.generate_paths(num_paths=1000)
        energy_fluctuation = compute_energy_fluctuation(paths)
        assert energy_fluctuation < self.tolerance

Links to:

• unit_testing
• integration_testing
• test_driven_development

Performance Benchmarks

class PathBenchmarks:
    """Benchmark suite for path methods."""
    
    def benchmark_sampling(self,
                         methods: List[Callable],
                         system: Callable) -> Dict[str, float]:
        """Benchmark different sampling methods."""
        results = {}
        for method in methods:
            start_time = time.time()
            paths = method(system)
            end_time = time.time()
            
            results[method.__name__] = {
                'time': end_time - start_time,
                'efficiency': compute_sampling_efficiency(paths)
            }
        return results

Links to:

• performance_testing
• benchmarking
• profiling

References

• neal_2011 - MCMC Using Hamiltonian Dynamics
• girolami_2011 - Riemann Manifold Langevin and Hamiltonian Monte Carlo
• hoffman_2014 - The No-U-Turn Sampler
• betancourt_2017 - A Conceptual Introduction to Hamiltonian Monte Carlo