Advanced Techniques and Applications¶
This tutorial covers advanced techniques for using Entropic AI in complex, real-world scenarios. We explore sophisticated applications that combine multiple thermodynamic principles, multi-scale optimization, and domain-specific adaptations.
Overview¶
Advanced applications of Entropic AI involve:
- Multi-Scale Thermodynamics: Handling systems with multiple temporal and spatial scales
- Adaptive Temperature Control: Dynamic temperature schedules based on evolution progress
- Hybrid Optimization: Combining thermodynamic evolution with other optimization paradigms
- Real-Time Evolution: Continuous adaptation in dynamic environments
- Quantum-Inspired Extensions: Leveraging quantum thermodynamic principles
Prerequisites¶
import numpy as np
import torch
import torch.nn as nn
from scipy.optimize import minimize
from entropic-ai.core import ThermodynamicNetwork, ComplexityOptimizer, GenerativeDiffuser
from entropic-ai.advanced import MultiScaleEvolver, AdaptiveThermostat, HybridOptimizer
from entropic-ai.quantum import QuantumThermodynamicNetwork
from entropic-ai.realtime import StreamingEvolver, DynamicAdaptation
Multi-Scale Thermodynamic Systems¶
Hierarchical Temperature Dynamics¶
Handle systems with multiple characteristic time scales:
class MultiScaleThermodynamicSystem:
def __init__(self, scale_hierarchy):
self.scale_hierarchy = scale_hierarchy
self.scale_networks = {}
self.scale_temperatures = {}
self.coupling_strengths = {}
self._initialize_scales()
def _initialize_scales(self):
"""Initialize networks and parameters for each scale."""
for scale_name, scale_config in self.scale_hierarchy.items():
# Create scale-specific network
self.scale_networks[scale_name] = ThermodynamicNetwork(
input_dim=scale_config['input_dim'],
hidden_dims=scale_config['hidden_dims'],
output_dim=scale_config['output_dim'],
temperature=scale_config['initial_temperature']
)
# Initialize scale temperature
self.scale_temperatures[scale_name] = scale_config['initial_temperature']
# Set coupling strengths to other scales
self.coupling_strengths[scale_name] = scale_config.get('coupling_strengths', {})
def evolve_multiscale(self, input_data, num_steps=1000):
"""Evolve system across all scales simultaneously."""
evolution_history = {scale: [] for scale in self.scale_hierarchy.keys()}
for step in range(num_steps):
# Update each scale
scale_updates = {}
for scale_name in self.scale_hierarchy.keys():
# Compute scale-specific forces
internal_force = self._compute_internal_force(scale_name, input_data)
coupling_force = self._compute_coupling_force(scale_name, scale_updates)
total_force = internal_force + coupling_force
# Update scale state
scale_update = self._update_scale_state(
scale_name,
total_force,
self.scale_temperatures[scale_name]
)
scale_updates[scale_name] = scale_update
evolution_history[scale_name].append(scale_update)
# Update temperatures according to scale-specific schedules
self._update_scale_temperatures(step)
# Check convergence across scales
if self._check_multiscale_convergence(scale_updates):
break
return evolution_history
def _compute_internal_force(self, scale_name, input_data):
"""Compute internal thermodynamic force for a scale."""
network = self.scale_networks[scale_name]
# Forward pass to get current state
current_state = network(input_data)
# Compute energy gradient
energy = self._compute_scale_energy(scale_name, current_state)
energy_gradient = torch.autograd.grad(energy, current_state, retain_graph=True)[0]
# Compute entropy gradient
entropy = self._compute_scale_entropy(scale_name, current_state)
entropy_gradient = torch.autograd.grad(entropy, current_state, retain_graph=True)[0]
# Thermodynamic force: F = -∇E + T∇S
temperature = self.scale_temperatures[scale_name]
force = -energy_gradient + temperature * entropy_gradient
return force
def _compute_coupling_force(self, scale_name, scale_updates):
"""Compute coupling force from other scales."""
coupling_force = torch.zeros_like(self.scale_networks[scale_name].get_state())
for other_scale, coupling_strength in self.coupling_strengths[scale_name].items():
if other_scale in scale_updates:
# Coupling force proportional to state difference
other_state = scale_updates[other_scale]
current_state = self.scale_networks[scale_name].get_state()
# Scale-dependent coupling (may need projection/interpolation)
projected_other_state = self._project_between_scales(
other_state, other_scale, scale_name
)
coupling_force += coupling_strength * (projected_other_state - current_state)
return coupling_force
def _project_between_scales(self, state, source_scale, target_scale):
"""Project state from source scale to target scale."""
source_config = self.scale_hierarchy[source_scale]
target_config = self.scale_hierarchy[target_scale]
# Simple linear projection (can be made more sophisticated)
if source_config['output_dim'] != target_config['output_dim']:
projection_matrix = torch.randn(
target_config['output_dim'],
source_config['output_dim']
)
projected_state = torch.matmul(projection_matrix, state)
else:
projected_state = state
return projected_state
Scale-Adaptive Evolution¶
Automatically adapt evolution parameters based on scale dynamics:
class ScaleAdaptiveEvolution:
def __init__(self, base_evolver):
self.base_evolver = base_evolver
self.scale_detectors = {
'temporal': TemporalScaleDetector(),
'spatial': SpatialScaleDetector(),
'complexity': ComplexityScaleDetector()
}
def evolve_with_scale_adaptation(self, initial_state, target_objective):
"""Evolve with automatic scale adaptation."""
current_state = initial_state
evolution_history = []
for iteration in range(self.max_iterations):
# Detect current system scales
detected_scales = self._detect_system_scales(current_state)
# Adapt evolution parameters based on scales
adapted_params = self._adapt_evolution_parameters(detected_scales)
# Update evolver with adapted parameters
self.base_evolver.update_parameters(adapted_params)
# Perform evolution step
next_state = self.base_evolver.evolution_step(current_state, target_objective)
# Record evolution
evolution_history.append({
'state': current_state,
'scales': detected_scales,
'parameters': adapted_params
})
current_state = next_state
# Check convergence
if self._check_convergence(current_state, target_objective):
break
return current_state, evolution_history
def _detect_system_scales(self, state):
"""Detect characteristic scales in current system state."""
scales = {}
for scale_type, detector in self.scale_detectors.items():
detected_scale = detector.detect(state)
scales[scale_type] = detected_scale
return scales
def _adapt_evolution_parameters(self, detected_scales):
"""Adapt evolution parameters based on detected scales."""
adapted_params = {}
# Adapt temperature based on temporal scale
temporal_scale = detected_scales['temporal']
if temporal_scale > 100: # Slow dynamics
adapted_params['temperature'] = self.base_evolver.temperature * 1.2
adapted_params['cooling_rate'] = 0.99
elif temporal_scale < 10: # Fast dynamics
adapted_params['temperature'] = self.base_evolver.temperature * 0.8
adapted_params['cooling_rate'] = 0.95
# Adapt complexity target based on complexity scale
complexity_scale = detected_scales['complexity']
adapted_params['target_complexity'] = min(0.9, max(0.1, complexity_scale))
return adapted_params
Adaptive Temperature Control¶
Reinforcement Learning-Based Thermostat¶
Use RL to learn optimal temperature schedules:
class RLThermostat:
def __init__(self, state_dim, action_dim):
self.state_dim = state_dim
self.action_dim = action_dim
# Neural network for Q-function
self.q_network = nn.Sequential(
nn.Linear(state_dim + action_dim, 128),
nn.ReLU(),
nn.Linear(128, 128),
nn.ReLU(),
nn.Linear(128, 1)
)
self.optimizer = torch.optim.Adam(self.q_network.parameters(), lr=1e-3)
self.experience_buffer = []
def select_temperature(self, evolution_state):
"""Select optimal temperature based on current evolution state."""
# Extract state features
state_features = self._extract_state_features(evolution_state)
# Epsilon-greedy action selection
if np.random.random() < self.epsilon:
# Random temperature
temperature = np.random.uniform(0.01, 10.0)
else:
# Greedy temperature selection
temperature = self._greedy_temperature_selection(state_features)
return temperature
def _extract_state_features(self, evolution_state):
"""Extract features from evolution state for RL."""
features = []
# Energy statistics
energy_mean = torch.mean(evolution_state.energies)
energy_std = torch.std(evolution_state.energies)
features.extend([energy_mean, energy_std])
# Entropy statistics
entropy_mean = torch.mean(evolution_state.entropies)
entropy_std = torch.std(evolution_state.entropies)
features.extend([entropy_mean, entropy_std])
# Population diversity
diversity = self._compute_population_diversity(evolution_state.population)
features.append(diversity)
# Convergence rate
convergence_rate = self._compute_convergence_rate(evolution_state.history)
features.append(convergence_rate)
# Acceptance rate
acceptance_rate = evolution_state.acceptance_rate
features.append(acceptance_rate)
return torch.tensor(features, dtype=torch.float32)
def _greedy_temperature_selection(self, state_features):
"""Select temperature that maximizes Q-value."""
best_temperature = 0.01
best_q_value = -float('inf')
# Search over temperature range
for temp in np.linspace(0.01, 10.0, 100):
# Create state-action pair
state_action = torch.cat([state_features, torch.tensor([temp])])
# Compute Q-value
q_value = self.q_network(state_action)
if q_value > best_q_value:
best_q_value = q_value
best_temperature = temp
return best_temperature
def update_q_network(self, experience_batch):
"""Update Q-network using experience batch."""
states, actions, rewards, next_states = experience_batch
# Current Q-values
current_q = self.q_network(torch.cat([states, actions], dim=1))
# Target Q-values (using target network or temporal difference)
with torch.no_grad():
next_q = self._compute_next_q_values(next_states)
target_q = rewards + self.gamma * next_q
# Compute loss
loss = nn.MSELoss()(current_q, target_q)
# Update network
self.optimizer.zero_grad()
loss.backward()
self.optimizer.step()
Adaptive Cooling Schedules¶
Implement intelligent cooling schedules:
class AdaptiveCoolingSchedule:
def __init__(self, initial_temperature, adaptation_rate=0.1):
self.initial_temperature = initial_temperature
self.current_temperature = initial_temperature
self.adaptation_rate = adaptation_rate
# History tracking
self.temperature_history = [initial_temperature]
self.acceptance_history = []
self.energy_history = []
def update_temperature(self, evolution_metrics):
"""Update temperature based on evolution metrics."""
acceptance_rate = evolution_metrics['acceptance_rate']
energy_improvement = evolution_metrics['energy_improvement']
convergence_rate = evolution_metrics['convergence_rate']
# Compute temperature adjustment
adjustment = self._compute_temperature_adjustment(
acceptance_rate, energy_improvement, convergence_rate
)
# Update temperature
self.current_temperature *= (1 + self.adaptation_rate * adjustment)
# Ensure temperature bounds
self.current_temperature = max(0.001, min(100.0, self.current_temperature))
# Record history
self.temperature_history.append(self.current_temperature)
self.acceptance_history.append(acceptance_rate)
self.energy_history.append(energy_improvement)
return self.current_temperature
def _compute_temperature_adjustment(self, acceptance_rate, energy_improvement, convergence_rate):
"""Compute temperature adjustment based on multiple metrics."""
adjustment = 0.0
# Acceptance rate feedback
target_acceptance = 0.44 # Optimal for many problems
if acceptance_rate < target_acceptance - 0.1:
adjustment += 0.1 # Increase temperature to increase acceptance
elif acceptance_rate > target_acceptance + 0.1:
adjustment -= 0.1 # Decrease temperature to decrease acceptance
# Energy improvement feedback
if energy_improvement < 0.001: # Slow improvement
adjustment += 0.05 # Increase temperature for exploration
elif energy_improvement > 0.1: # Fast improvement
adjustment -= 0.05 # Decrease temperature for exploitation
# Convergence rate feedback
if convergence_rate < 0.001: # Slow convergence
adjustment += 0.02 # Increase temperature
elif convergence_rate > 0.1: # Fast convergence (might be premature)
adjustment += 0.01 # Slightly increase temperature
return adjustment
Hybrid Optimization Approaches¶
Thermodynamic-Gradient Hybrid¶
Combine thermodynamic evolution with gradient-based optimization:
class ThermodynamicGradientHybrid:
def __init__(self, thermodynamic_evolver, gradient_optimizer):
self.thermodynamic_evolver = thermodynamic_evolver
self.gradient_optimizer = gradient_optimizer
self.switching_strategy = 'adaptive'
def hybrid_optimize(self, objective_function, initial_state):
"""Hybrid optimization using both approaches."""
current_state = initial_state
optimization_history = []
for iteration in range(self.max_iterations):
# Decide which optimizer to use
optimizer_choice = self._select_optimizer(current_state, iteration)
if optimizer_choice == 'thermodynamic':
# Use thermodynamic evolution
next_state = self.thermodynamic_evolver.evolve_step(
current_state, objective_function
)
method_used = 'thermodynamic'
else:
# Use gradient-based optimization
next_state = self.gradient_optimizer.optimize_step(
current_state, objective_function
)
method_used = 'gradient'
# Record optimization step
optimization_history.append({
'iteration': iteration,
'state': current_state,
'objective_value': objective_function(current_state),
'method': method_used
})
current_state = next_state
# Check convergence
if self._check_convergence(current_state, objective_function):
break
return current_state, optimization_history
def _select_optimizer(self, current_state, iteration):
"""Select which optimizer to use based on current conditions."""
if self.switching_strategy == 'adaptive':
# Adaptive switching based on landscape characteristics
landscape_roughness = self._estimate_landscape_roughness(current_state)
gradient_norm = self._estimate_gradient_norm(current_state)
if landscape_roughness > 0.5 or gradient_norm < 0.01:
return 'thermodynamic' # Use for rough landscapes or weak gradients
else:
return 'gradient' # Use for smooth landscapes with strong gradients
elif self.switching_strategy == 'alternating':
# Simple alternating strategy
return 'thermodynamic' if iteration % 2 == 0 else 'gradient'
elif self.switching_strategy == 'phased':
# Phase-based strategy: start with thermodynamic, switch to gradient
if iteration < self.max_iterations // 2:
return 'thermodynamic'
else:
return 'gradient'
def _estimate_landscape_roughness(self, state):
"""Estimate local landscape roughness."""
# Sample nearby points
perturbations = [torch.randn_like(state) * 0.01 for _ in range(10)]
nearby_states = [state + perturbation for perturbation in perturbations]
# Evaluate objective at nearby points
nearby_values = [self.objective_function(nearby_state) for nearby_state in nearby_states]
current_value = self.objective_function(state)
# Compute roughness as variance of nearby values
value_variance = np.var(nearby_values + [current_value])
return value_variance
Multi-Objective Thermodynamic Optimization¶
Handle multiple competing objectives:
class MultiObjectiveThermodynamicOptimizer:
def __init__(self, objectives, weights=None):
self.objectives = objectives
self.num_objectives = len(objectives)
if weights is None:
self.weights = [1.0 / self.num_objectives] * self.num_objectives
else:
self.weights = weights
self.pareto_front = []
def multi_objective_evolve(self, initial_population):
"""Evolve population for multi-objective optimization."""
population = initial_population
evolution_history = []
for generation in range(self.max_generations):
# Evaluate all objectives for population
objective_values = self._evaluate_population_objectives(population)
# Update Pareto front
self._update_pareto_front(population, objective_values)
# Compute multi-objective fitness
fitness_values = self._compute_multi_objective_fitness(objective_values)
# Thermodynamic selection and reproduction
new_population = self._thermodynamic_selection_reproduction(
population, fitness_values
)
population = new_population
# Record evolution
evolution_history.append({
'generation': generation,
'population': population.copy(),
'objective_values': objective_values,
'pareto_front': self.pareto_front.copy()
})
return self.pareto_front, evolution_history
def _evaluate_population_objectives(self, population):
"""Evaluate all objectives for population members."""
objective_values = []
for individual in population:
individual_objectives = []
for objective in self.objectives:
obj_value = objective(individual)
individual_objectives.append(obj_value)
objective_values.append(individual_objectives)
return objective_values
def _compute_multi_objective_fitness(self, objective_values):
"""Compute multi-objective fitness using thermodynamic principles."""
fitness_values = []
for individual_objectives in objective_values:
# Weighted scalarization
weighted_sum = sum(
w * obj for w, obj in zip(self.weights, individual_objectives)
)
# Pareto dominance bonus
dominance_bonus = self._compute_dominance_bonus(individual_objectives)
# Diversity bonus (entropy)
diversity_bonus = self._compute_diversity_bonus(individual_objectives)
# Total fitness (energy)
total_fitness = weighted_sum - dominance_bonus - diversity_bonus
fitness_values.append(total_fitness)
return fitness_values
def _update_pareto_front(self, population, objective_values):
"""Update Pareto front with non-dominated solutions."""
# Combine current population with existing Pareto front
all_individuals = list(population) + [sol['individual'] for sol in self.pareto_front]
all_objectives = list(objective_values) + [sol['objectives'] for sol in self.pareto_front]
# Find non-dominated solutions
non_dominated = []
for i, (individual, objectives) in enumerate(zip(all_individuals, all_objectives)):
is_dominated = False
for j, other_objectives in enumerate(all_objectives):
if i != j and self._dominates(other_objectives, objectives):
is_dominated = True
break
if not is_dominated:
non_dominated.append({
'individual': individual,
'objectives': objectives
})
self.pareto_front = non_dominated
def _dominates(self, objectives1, objectives2):
"""Check if objectives1 dominates objectives2."""
# Assumes minimization
better_in_all = all(obj1 <= obj2 for obj1, obj2 in zip(objectives1, objectives2))
better_in_at_least_one = any(obj1 < obj2 for obj1, obj2 in zip(objectives1, objectives2))
return better_in_all and better_in_at_least_one
Real-Time Evolutionary Systems¶
Streaming Data Evolution¶
Handle continuously arriving data:
class StreamingThermodynamicEvolver:
def __init__(self, base_evolver, adaptation_rate=0.1):
self.base_evolver = base_evolver
self.adaptation_rate = adaptation_rate
# Streaming state
self.current_model = None
self.streaming_buffer = []
self.adaptation_triggers = {
'data_drift': DataDriftDetector(),
'performance_drop': PerformanceMonitor(),
'concept_shift': ConceptShiftDetector()
}
def process_streaming_data(self, data_stream):
"""Process streaming data with continuous evolution."""
for data_batch in data_stream:
# Add to streaming buffer
self.streaming_buffer.append(data_batch)
# Check adaptation triggers
should_adapt = self._check_adaptation_triggers(data_batch)
if should_adapt:
# Trigger thermodynamic evolution
self._adapt_model(data_batch)
# Clear adaptation triggers
self._reset_adaptation_triggers()
# Process current batch
predictions = self.current_model.predict(data_batch)
# Update streaming buffer (maintain size limit)
if len(self.streaming_buffer) > self.max_buffer_size:
self.streaming_buffer.pop(0)
yield predictions
def _check_adaptation_triggers(self, data_batch):
"""Check if adaptation should be triggered."""
triggers_fired = []
for trigger_name, trigger in self.adaptation_triggers.items():
if trigger.should_trigger(data_batch, self.current_model):
triggers_fired.append(trigger_name)
# Adapt if any trigger fires
return len(triggers_fired) > 0
def _adapt_model(self, trigger_data):
"""Adapt model using thermodynamic evolution."""
# Prepare evolution data
evolution_data = self._prepare_evolution_data()
# Set evolution objective based on current performance
objective = self._create_adaptive_objective(trigger_data)
# Run thermodynamic evolution
evolved_model = self.base_evolver.evolve(
initial_state=self.current_model,
objective_function=objective,
evolution_data=evolution_data
)
# Update current model
self.current_model = evolved_model
def _prepare_evolution_data(self):
"""Prepare data for evolution from streaming buffer."""
# Use recent data for evolution
recent_data = self.streaming_buffer[-self.evolution_window_size:]
# Combine and preprocess
evolution_data = self._combine_data_batches(recent_data)
return evolution_data
def _create_adaptive_objective(self, trigger_data):
"""Create objective function adapted to current conditions."""
def adaptive_objective(model):
# Base performance on trigger data
base_performance = self._evaluate_model_performance(model, trigger_data)
# Add adaptation penalties/bonuses
adaptation_penalty = self._compute_adaptation_penalty(model)
stability_bonus = self._compute_stability_bonus(model)
return base_performance + adaptation_penalty - stability_bonus
return adaptive_objective
Dynamic Environment Adaptation¶
Adapt to changing environments:
class DynamicEnvironmentAdapter:
def __init__(self, environment_monitor):
self.environment_monitor = environment_monitor
self.adaptation_history = []
self.environment_models = {}
def adapt_to_environment_changes(self, base_system):
"""Continuously adapt system to environment changes."""
current_system = base_system
while self.environment_monitor.is_active():
# Monitor environment
environment_state = self.environment_monitor.get_current_state()
# Detect environment changes
environment_change = self._detect_environment_change(environment_state)
if environment_change:
# Adapt system to new environment
adapted_system = self._adapt_system_to_environment(
current_system, environment_state
)
# Record adaptation
self.adaptation_history.append({
'timestamp': self.environment_monitor.get_timestamp(),
'environment_state': environment_state,
'system_before': current_system,
'system_after': adapted_system,
'adaptation_method': 'thermodynamic_evolution'
})
current_system = adapted_system
# Wait for next monitoring cycle
time.sleep(self.monitoring_interval)
return current_system, self.adaptation_history
def _adapt_system_to_environment(self, system, environment_state):
"""Adapt system to specific environment state."""
# Check if we have a model for this environment
env_signature = self._compute_environment_signature(environment_state)
if env_signature in self.environment_models:
# Use existing environment model
environment_model = self.environment_models[env_signature]
else:
# Create new environment model
environment_model = self._create_environment_model(environment_state)
self.environment_models[env_signature] = environment_model
# Adapt system using environment model
adapted_system = self._thermodynamic_adaptation(system, environment_model)
return adapted_system
def _thermodynamic_adaptation(self, system, environment_model):
"""Perform thermodynamic adaptation to environment."""
# Create environment-aware energy function
def environment_energy(system_state):
# Base system energy
base_energy = system.compute_energy(system_state)
# Environment interaction energy
interaction_energy = environment_model.compute_interaction_energy(
system_state, self.environment_monitor.get_current_state()
)
return base_energy + interaction_energy
# Create environment-aware entropy function
def environment_entropy(system_state):
# Base system entropy
base_entropy = system.compute_entropy(system_state)
# Environment diversity bonus
diversity_bonus = environment_model.compute_diversity_bonus(system_state)
return base_entropy + diversity_bonus
# Run thermodynamic evolution with environment awareness
evolver = ThermodynamicEvolver(
energy_function=environment_energy,
entropy_function=environment_entropy,
temperature_schedule='adaptive'
)
adapted_system = evolver.evolve(system)
return adapted_system
Quantum-Inspired Extensions¶
Quantum Thermodynamic Networks¶
Incorporate quantum mechanical principles:
class QuantumThermodynamicNetwork:
def __init__(self, num_qubits, temperature):
self.num_qubits = num_qubits
self.temperature = temperature
self.quantum_state = self._initialize_quantum_state()
def _initialize_quantum_state(self):
"""Initialize quantum state in thermal equilibrium."""
# Create density matrix for thermal state
# ρ = exp(-βH) / Tr(exp(-βH))
beta = 1.0 / self.temperature
# Simple Hamiltonian (can be made more complex)
hamiltonian = self._create_hamiltonian()
# Compute thermal state
thermal_state = torch.matrix_exp(-beta * hamiltonian)
thermal_state = thermal_state / torch.trace(thermal_state)
return thermal_state
def _create_hamiltonian(self):
"""Create system Hamiltonian."""
# Example: Ising-like Hamiltonian
dim = 2 ** self.num_qubits
hamiltonian = torch.zeros(dim, dim, dtype=torch.complex64)
# Add terms to Hamiltonian
for i in range(self.num_qubits):
# Local field terms
local_term = self._create_pauli_z_term(i)
hamiltonian += local_term
# Interaction terms
if i < self.num_qubits - 1:
interaction_term = self._create_interaction_term(i, i+1)
hamiltonian += interaction_term
return hamiltonian
def quantum_evolution_step(self, external_field):
"""Perform quantum thermodynamic evolution step."""
# Time evolution operator
dt = 0.01
evolution_hamiltonian = self._create_hamiltonian() + external_field
evolution_operator = torch.matrix_exp(-1j * dt * evolution_hamiltonian)
# Apply evolution
self.quantum_state = torch.matmul(
torch.matmul(evolution_operator, self.quantum_state),
torch.conj(evolution_operator.T)
)
# Apply thermalization
self._apply_thermalization()
return self.quantum_state
def _apply_thermalization(self):
"""Apply thermalization to quantum state."""
# Lindblad master equation approach
# Simplified: mix with thermal state
beta = 1.0 / self.temperature
thermal_state = self._compute_thermal_state(beta)
# Mixing parameter (thermalization rate)
gamma = 0.01
self.quantum_state = (
(1 - gamma) * self.quantum_state +
gamma * thermal_state
)
def measure_quantum_observables(self):
"""Measure quantum observables."""
observables = {}
# Energy expectation value
hamiltonian = self._create_hamiltonian()
energy = torch.trace(torch.matmul(hamiltonian, self.quantum_state)).real
observables['energy'] = energy
# Von Neumann entropy
eigenvalues = torch.linalg.eigvals(self.quantum_state).real
eigenvalues = eigenvalues[eigenvalues > 1e-12] # Remove numerical zeros
entropy = -torch.sum(eigenvalues * torch.log(eigenvalues))
observables['entropy'] = entropy
# Quantum coherence measures
coherence = self._compute_quantum_coherence()
observables['coherence'] = coherence
return observables
Performance Optimization¶
GPU-Accelerated Evolution¶
Optimize for GPU computation:
class GPUAcceleratedEvolution:
def __init__(self, device='cuda'):
self.device = device
self.batch_processing = True
def parallel_population_evolution(self, population, objective_function):
"""Evolve entire population in parallel on GPU."""
# Convert population to GPU tensors
population_tensor = torch.stack(population).to(self.device)
batch_size = population_tensor.shape[0]
# Batch evaluate objectives
with torch.no_grad():
objective_values = self._batch_evaluate_objectives(
population_tensor, objective_function
)
# Batch compute thermodynamic forces
forces = self._batch_compute_forces(population_tensor, objective_values)
# Batch update population
new_population_tensor = self._batch_update_population(
population_tensor, forces
)
# Convert back to list
new_population = [new_population_tensor[i] for i in range(batch_size)]
return new_population
def _batch_evaluate_objectives(self, population_tensor, objective_function):
"""Evaluate objectives for entire population batch."""
# Vectorized objective evaluation
objective_values = torch.zeros(population_tensor.shape[0], device=self.device)
# Check if objective function supports batch evaluation
if hasattr(objective_function, 'batch_evaluate'):
objective_values = objective_function.batch_evaluate(population_tensor)
else:
# Fallback to individual evaluation
for i in range(population_tensor.shape[0]):
objective_values[i] = objective_function(population_tensor[i])
return objective_values
def _batch_compute_forces(self, population_tensor, objective_values):
"""Compute thermodynamic forces for entire population batch."""
# Enable gradient computation
population_tensor.requires_grad_(True)
# Compute energy gradients
energy_gradients = torch.autograd.grad(
objective_values.sum(), population_tensor,
create_graph=True, retain_graph=True
)[0]
# Compute entropy gradients (simplified)
entropy_values = self._batch_compute_entropy(population_tensor)
entropy_gradients = torch.autograd.grad(
entropy_values.sum(), population_tensor,
create_graph=True, retain_graph=True
)[0]
# Thermodynamic forces: F = -∇E + T∇S
forces = -energy_gradients + self.temperature * entropy_gradients
return forces
Best Practices for Advanced Applications¶
1. System Design¶
- Modular Architecture: Design systems with interchangeable components
- Scale Separation: Clearly separate different time and space scales
- Resource Management: Monitor and manage computational resources
2. Parameter Tuning¶
- Adaptive Parameters: Use adaptive strategies rather than fixed parameters
- Cross-Validation: Validate parameter choices across different scenarios
- Sensitivity Analysis: Understand parameter sensitivity
3. Performance Monitoring¶
- Real-Time Metrics: Monitor evolution progress in real-time
- Resource Utilization: Track CPU/GPU/memory usage
- Convergence Analysis: Analyze convergence patterns
4. Robustness¶
- Error Handling: Implement robust error handling and recovery
- Numerical Stability: Ensure numerical stability across platforms
- Fallback Strategies: Have fallback strategies for edge cases
This tutorial demonstrates how to push the boundaries of Entropic AI through advanced techniques that combine multiple optimization paradigms, handle complex multi-scale systems, and adapt to dynamic environments in real-time.