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Sim-to-Real Transfer Techniques

Overview​

Sim-to-real transfer, also known as reality gap bridging, is the process of transferring knowledge, policies, or behaviors learned in simulation to real-world robotic systems. This challenge arises because simulations, despite their sophistication, cannot perfectly model all aspects of the real world. This lesson explores the fundamental challenges of sim-to-real transfer and presents various techniques to bridge the reality gap, including domain randomization, domain adaptation, and other advanced approaches.

The Reality Gap Problem​

Understanding the Sim-to-Real Transfer Challenge​

The reality gap refers to the differences between simulation and real-world environments that can cause algorithms trained in simulation to fail when deployed on real robots. These differences manifest in several forms:

import numpy as np
import matplotlib.pyplot as plt

class RealityGapAnalyzer:
    def __init__(self):
        self.sim_params = {
            'friction': 0.0,  # Idealized friction
            'mass': 1.0,      # Perfectly known mass
            'geometry': 'perfect',  # Perfect geometric models
            'sensor_noise': 0.0,    # Noise-free sensors
            'actuator_dynamics': 'ideal'  # Instantaneous response
        }

        self.real_params = {
            'friction': 0.1,  # Real-world friction
            'mass': 1.05,     # Actual mass with variations
            'geometry': 'approximate',  # Approximated geometry
            'sensor_noise': 0.05,    # Real sensor noise
            'actuator_dynamics': 'delayed'  # Delayed response
        }

    def calculate_reality_gap(self, metric='euclidean'):
        """
        Calculate the reality gap between simulation and real parameters
        """
        gap = 0.0

        # Normalize and compare parameters
        friction_gap = abs(self.sim_params['friction'] - self.real_params['friction'])
        mass_gap = abs(self.sim_params['mass'] - self.real_params['mass'])
        noise_gap = abs(self.sim_params['sensor_noise'] - self.real_params['sensor_noise'])

        if metric == 'euclidean':
            gap = np.sqrt(friction_gap**2 + mass_gap**2 + noise_gap**2)
        elif metric == 'manhattan':
            gap = friction_gap + mass_gap + noise_gap
        elif metric == 'max':
            gap = max(friction_gap, mass_gap, noise_gap)

        return gap

    def analyze_transfer_failure_modes(self):
        """
        Analyze different failure modes in sim-to-real transfer
        """
        failure_modes = {
            'dynamics_mismatch': {
                'description': 'Simulated vs real robot dynamics differ',
                'impact': 'Control policies fail to achieve desired behavior',
                'severity': 'high',
                'mitigation': 'System identification, adaptive control'
            },
            'visual_appearance': {
                'description': 'Simulated vs real visual appearance differ',
                'impact': 'Computer vision algorithms fail on real images',
                'severity': 'high',
                'mitigation': 'Domain randomization, synthetic-to-real techniques'
            },
            'sensor_characteristics': {
                'description': 'Simulated vs real sensor properties differ',
                'impact': 'Perception and localization fail',
                'severity': 'high',
                'mitigation': 'Sensor calibration, noise modeling'
            },
            'environment_modeling': {
                'description': 'Simulated vs real environment differ',
                'impact': 'Navigation and manipulation fail',
                'severity': 'medium',
                'mitigation': 'Environment randomization, online learning'
            },
            'actuator_dynamics': {
                'description': 'Simulated vs real actuator behavior differ',
                'impact': 'Precise control becomes impossible',
                'severity': 'high',
                'mitigation': 'Detailed actuator modeling, feedback control'
            }
        }

        return failure_modes

    def plot_reality_gap_distribution(self):
        """
        Visualize the distribution of reality gaps across different parameters
        """
        parameters = list(self.sim_params.keys())
        sim_values = list(self.sim_params.values())
        real_values = list(self.real_params.values())

        x = np.arange(len(parameters))
        width = 0.35

        fig, ax = plt.subplots(figsize=(12, 6))
        ax.bar(x - width/2, sim_values, width, label='Simulation', alpha=0.8)
        ax.bar(x + width/2, real_values, width, label='Reality', alpha=0.8)

        ax.set_xlabel('Parameters')
        ax.set_ylabel('Values')
        ax.set_title('Reality Gap: Simulation vs Reality Parameter Comparison')
        ax.set_xticks(x)
        ax.set_xticklabels(parameters, rotation=45, ha='right')
        ax.legend()

        plt.tight_layout()
        plt.show()

        return fig

Quantifying the Transfer Gap​

class TransferGapQuantifier:
    def __init__(self):
        self.sim_performance = 0.0
        self.real_performance = 0.0
        self.transfer_efficiency = 0.0

    def calculate_transfer_metrics(self, sim_success_rate, real_success_rate):
        """
        Calculate various transfer metrics
        """
        metrics = {}

        # Absolute Performance Gap
        metrics['absolute_gap'] = sim_success_rate - real_success_rate

        # Relative Performance Gap
        if sim_success_rate > 0:
            metrics['relative_gap'] = (sim_success_rate - real_success_rate) / sim_success_rate
        else:
            metrics['relative_gap'] = float('inf')

        # Transfer Efficiency
        metrics['transfer_efficiency'] = real_success_rate / sim_success_rate if sim_success_rate > 0 else 0

        # Zero-Shot Success Rate
        metrics['zero_shot_success'] = real_success_rate

        # Sample Efficiency Ratio
        metrics['sample_efficiency'] = self.calculate_sample_efficiency(sim_success_rate, real_success_rate)

        return metrics

    def calculate_sample_efficiency(self, sim_samples, real_samples):
        """
        Calculate how efficiently samples translate from sim to real
        """
        # This is a simplified example
        # Real implementation would involve more complex analysis
        return real_samples / sim_samples if sim_samples > 0 else 0

    def calculate_domain_similarity(self, sim_features, real_features):
        """
        Calculate domain similarity using Maximum Mean Discrepancy (MMD)
        """
        # Simplified MMD calculation
        # In practice, this would use kernel methods
        mean_sim = np.mean(sim_features, axis=0)
        mean_real = np.mean(real_features, axis=0)

        # Euclidean distance between means as proxy for MMD
        mmd = np.linalg.norm(mean_sim - mean_real)

        return mmd

    def evaluate_transfer_robustness(self, policies, test_conditions):
        """
        Evaluate how robust policies are to condition changes
        """
        robustness_scores = []

        for policy in policies:
            condition_scores = []
            for condition in test_conditions:
                score = self.evaluate_policy_under_condition(policy, condition)
                condition_scores.append(score)

            # Robustness is the consistency across conditions
            robustness = 1.0 - np.std(condition_scores) / (np.mean(condition_scores) + 1e-8)
            robustness_scores.append(robustness)

        return robustness_scores

    def evaluate_policy_under_condition(self, policy, condition):
        """
        Evaluate policy performance under specific condition
        """
        # This would involve running the policy in simulation with the condition
        # and measuring performance
        return np.random.random()  # Placeholder

Domain Randomization Techniques​

Basic Domain Randomization​

Domain randomization involves randomizing simulation parameters to make policies robust to variations:

import numpy as np
import random

class DomainRandomizer:
    def __init__(self):
        self.randomization_ranges = {
            'friction': (0.0, 1.0),
            'mass': (0.8, 1.2),
            'restitution': (0.0, 0.5),
            'gravity': (9.5, 10.1),
            'texture_resolution': (100, 1000),
            'lighting_condition': (0.5, 2.0),
            'camera_noise': (0.0, 0.1),
            'actuator_delay': (0.0, 0.05)
        }

        self.current_params = {}
        self.reset()

    def reset(self):
        """Reset to random domain parameters"""
        self.current_params = {}
        for param, (min_val, max_val) in self.randomization_ranges.items():
            self.current_params[param] = random.uniform(min_val, max_val)

    def randomize_environment(self):
        """Apply randomization to simulation environment"""
        randomized_env = {}

        # Randomize physical properties
        randomized_env['friction'] = self.current_params['friction']
        randomized_env['mass_multiplier'] = self.current_params['mass']
        randomized_env['restitution'] = self.current_params['restitution']
        randomized_env['gravity'] = self.current_params['gravity']

        # Randomize visual properties
        randomized_env['texture_complexity'] = self.current_params['texture_resolution']
        randomized_env['lighting_intensity'] = self.current_params['lighting_condition']

        # Randomize sensor properties
        randomized_env['camera_noise_level'] = self.current_params['camera_noise']
        randomized_env['actuator_response_time'] = self.current_params['actuator_delay']

        return randomized_env

    def update_randomization_ranges(self, new_ranges):
        """Update the randomization ranges based on real-world observations"""
        for param, (min_val, max_val) in new_ranges.items():
            if param in self.randomization_ranges:
                self.randomization_ranges[param] = (min_val, max_val)

    def randomize_episode(self):
        """Randomize for a new episode"""
        self.reset()
        return self.randomize_environment()

    def get_current_domain_params(self):
        """Get current domain parameters"""
        return self.current_params.copy()

Advanced Domain Randomization​

class AdvancedDomainRandomizer:
    def __init__(self):
        self.param_groups = {
            'dynamics': ['friction', 'mass', 'restitution', 'gravity'],
            'visual': ['texture_resolution', 'lighting_condition', 'camera_noise'],
            'actuation': ['actuator_delay', 'max_force', 'gear_ratio']
        }

        self.correlation_matrix = self.initialize_correlation_matrix()
        self.randomization_schedule = self.create_adaptive_schedule()

    def initialize_correlation_matrix(self):
        """
        Initialize correlation matrix for correlated parameter randomization
        """
        n_params = len(self.get_all_param_names())
        corr_matrix = np.eye(n_params)

        # Define correlations between related parameters
        # Example: friction and restitution might be correlated
        param_indices = {name: idx for idx, name in enumerate(self.get_all_param_names())}

        # Add some correlations (these would be learned from data in practice)
        friction_idx = param_indices.get('friction', -1)
        restitution_idx = param_indices.get('restitution', -1)

        if friction_idx >= 0 and restitution_idx >= 0:
            corr_matrix[friction_idx, restitution_idx] = 0.3
            corr_matrix[restitution_idx, friction_idx] = 0.3

        return corr_matrix

    def get_all_param_names(self):
        """Get all parameter names"""
        all_names = []
        for group in self.param_groups.values():
            all_names.extend(group)
        return all_names

    def randomize_with_correlations(self):
        """
        Randomize parameters considering correlations
        """
        param_names = self.get_all_param_names()
        n_params = len(param_names)

        # Generate correlated random variables using Cholesky decomposition
        L = np.linalg.cholesky(self.correlation_matrix)
        uncorrelated_vars = np.random.randn(n_params)
        correlated_vars = L @ uncorrelated_vars

        randomized_params = {}
        for i, param_name in enumerate(param_names):
            # Use the correlated variable to influence the randomization
            base_min, base_max = self.get_base_range(param_name)
            correlation_influence = correlated_vars[i] * 0.1  # Influence factor

            # Adjust range based on correlation
            adjusted_min = max(base_min, base_min + correlation_influence)
            adjusted_max = min(base_max, base_max + correlation_influence)

            randomized_params[param_name] = np.random.uniform(adjusted_min, adjusted_max)

        return randomized_params

    def get_base_range(self, param_name):
        """Get base randomization range for parameter"""
        # This would typically come from a configuration
        ranges = {
            'friction': (0.0, 1.0),
            'mass': (0.8, 1.2),
            'restitution': (0.0, 0.5),
            'gravity': (9.5, 10.1),
            'texture_resolution': (100, 1000),
            'lighting_condition': (0.5, 2.0),
            'camera_noise': (0.0, 0.1),
            'actuator_delay': (0.0, 0.05),
            'max_force': (0.8, 1.2),
            'gear_ratio': (0.9, 1.1)
        }
        return ranges.get(param_name, (0.0, 1.0))

    def create_adaptive_schedule(self):
        """
        Create adaptive randomization schedule
        """
        schedule = {
            'early_phase': {
                'range_expansion_rate': 0.1,
                'randomization_frequency': 1,  # Every episode
                'focus_areas': ['dynamics']  # Start with dynamics
            },
            'mid_phase': {
                'range_expansion_rate': 0.05,
                'randomization_frequency': 1,
                'focus_areas': ['dynamics', 'visual']
            },
            'late_phase': {
                'range_expansion_rate': 0.02,
                'randomization_frequency': 2,  # Every 2 episodes
                'focus_areas': ['all']
            }
        }
        return schedule

    def adapt_randomization(self, performance_feedback):
        """
        Adapt randomization based on performance feedback
        """
        # This would analyze performance and adjust randomization strategy
        # For example, if performance is degrading, might narrow ranges temporarily
        # or focus on specific parameter groups
        pass

Curriculum Domain Randomization​

class CurriculumDomainRandomizer:
    def __init__(self):
        self.curriculum_levels = [
            {
                'name': 'easy',
                'difficulty': 0.1,
                'param_ranges': self.get_easy_ranges(),
                'duration': 1000,  # episodes
                'success_threshold': 0.8
            },
            {
                'name': 'medium',
                'difficulty': 0.3,
                'param_ranges': self.get_medium_ranges(),
                'duration': 2000,
                'success_threshold': 0.7
            },
            {
                'name': 'hard',
                'difficulty': 0.6,
                'param_ranges': self.get_hard_ranges(),
                'duration': 3000,
                'success_threshold': 0.6
            },
            {
                'name': 'extreme',
                'difficulty': 1.0,
                'param_ranges': self.get_extreme_ranges(),
                'duration': float('inf'),  # Continue indefinitely
                'success_threshold': 0.5
            }
        ]

        self.current_level = 0
        self.episode_count = 0
        self.performance_history = []

    def get_easy_ranges(self):
        """Get parameter ranges for easy level"""
        return {
            'friction': (0.05, 0.15),
            'mass': (0.95, 1.05),
            'restitution': (0.05, 0.15),
            'gravity': (9.7, 9.9),
            'camera_noise': (0.0, 0.02)
        }

    def get_medium_ranges(self):
        """Get parameter ranges for medium level"""
        return {
            'friction': (0.0, 0.3),
            'mass': (0.9, 1.1),
            'restitution': (0.0, 0.25),
            'gravity': (9.6, 10.0),
            'camera_noise': (0.0, 0.05)
        }

    def get_hard_ranges(self):
        """Get parameter ranges for hard level"""
        return {
            'friction': (0.0, 0.6),
            'mass': (0.8, 1.2),
            'restitution': (0.0, 0.4),
            'gravity': (9.5, 10.1),
            'camera_noise': (0.0, 0.08)
        }

    def get_extreme_ranges(self):
        """Get parameter ranges for extreme level"""
        return {
            'friction': (0.0, 1.0),
            'mass': (0.7, 1.3),
            'restitution': (0.0, 0.5),
            'gravity': (9.3, 10.3),
            'camera_noise': (0.0, 0.1)
        }

    def get_current_randomization(self):
        """Get current randomization parameters"""
        current_ranges = self.curriculum_levels[self.current_level]['param_ranges']

        randomized_params = {}
        for param, (min_val, max_val) in current_ranges.items():
            randomized_params[param] = random.uniform(min_val, max_val)

        return randomized_params

    def update_curriculum(self, episode_performance):
        """Update curriculum level based on performance"""
        self.performance_history.append(episode_performance)
        self.episode_count += 1

        # Check if we should advance to next level
        if (self.episode_count >= self.curriculum_levels[self.current_level]['duration'] and
            self.get_recent_performance() >= self.curriculum_levels[self.current_level]['success_threshold']):

            if self.current_level < len(self.curriculum_levels) - 1:
                self.current_level += 1
                self.episode_count = 0  # Reset counter for new level
                print(f"Advancing to curriculum level: {self.curriculum_levels[self.current_level]['name']}")

    def get_recent_performance(self, window=100):
        """Get average performance over recent episodes"""
        recent = self.performance_history[-window:]
        if not recent:
            return 0.0
        return sum(recent) / len(recent)

    def get_difficulty_level(self):
        """Get current difficulty level"""
        return self.curriculum_levels[self.current_level]['name']

Domain Adaptation Techniques​

Unsupervised Domain Adaptation​

import torch
import torch.nn as nn
import torch.nn.functional as F

class DomainAdaptationNetwork(nn.Module):
    def __init__(self, input_dim, hidden_dim=256, num_classes=10):
        super(DomainAdaptationNetwork, self).__init__()

        # Feature extractor
        self.feature_extractor = nn.Sequential(
            nn.Linear(input_dim, hidden_dim),
            nn.ReLU(),
            nn.Dropout(0.5),
            nn.Linear(hidden_dim, hidden_dim),
            nn.ReLU(),
            nn.Dropout(0.5),
            nn.Linear(hidden_dim, hidden_dim // 2),
            nn.ReLU()
        )

        # Label classifier
        self.label_classifier = nn.Sequential(
            nn.Linear(hidden_dim // 2, hidden_dim // 4),
            nn.ReLU(),
            nn.Linear(hidden_dim // 4, num_classes)
        )

        # Domain classifier
        self.domain_classifier = nn.Sequential(
            nn.Linear(hidden_dim // 2, hidden_dim // 4),
            nn.ReLU(),
            nn.Linear(hidden_dim // 4, 1),
            nn.Sigmoid()
        )

    def forward(self, x, lambda_val=0.0):
        features = self.feature_extractor(x)

        # Reverse gradient for domain adaptation
        reversed_features = GradientReverseLayer(lambda_val)(features)

        class_output = self.label_classifier(features)
        domain_output = self.domain_classifier(reversed_features)

        return class_output, domain_output

class GradientReverseLayer(torch.autograd.Function):
    """
    Gradient Reversal Layer
    """
    @staticmethod
    def forward(ctx, input, lambda_val):
        ctx.lambda_val = lambda_val
        return input

    @staticmethod
    def backward(ctx, grad_output):
        lambda_val = ctx.lambda_val
        return -lambda_val * grad_output, None

class DomainAdaptationTrainer:
    def __init__(self, model, learning_rate=0.001):
        self.model = model
        self.optimizer = torch.optim.Adam(model.parameters(), lr=learning_rate)
        self.class_criterion = nn.CrossEntropyLoss()
        self.domain_criterion = nn.BCELoss()

    def train_step(self, sim_data, real_data, sim_labels, real_labels=None):
        """
        Training step for domain adaptation
        """
        batch_size = sim_data.size(0)

        # Create domain labels (0 for sim, 1 for real)
        sim_domain_labels = torch.zeros(batch_size).to(sim_data.device)
        real_domain_labels = torch.ones(batch_size).to(real_data.device)

        # Concatenate data
        all_data = torch.cat([sim_data, real_data], dim=0)
        all_domain_labels = torch.cat([sim_domain_labels, real_domain_labels], dim=0)

        self.optimizer.zero_grad()

        # Forward pass
        class_outputs, domain_outputs = self.model(all_data, lambda_val=1.0)

        # Split outputs
        sim_class_out = class_outputs[:batch_size]
        real_class_out = class_outputs[batch_size:]
        domain_out = domain_outputs.squeeze()

        # Compute losses
        # Classification loss on simulated data (and real if labels available)
        class_loss = self.class_criterion(sim_class_out, sim_labels)
        if real_labels is not None:
            class_loss += self.class_criterion(real_class_out, real_labels)

        # Domain confusion loss
        domain_loss = self.domain_criterion(domain_out, all_domain_labels)

        # Total loss
        total_loss = class_loss + domain_loss

        total_loss.backward()
        self.optimizer.step()

        return total_loss.item(), class_loss.item(), domain_loss.item()

Adversarial Domain Adaptation​

class AdversarialDomainAdapter:
    def __init__(self, feature_dim, hidden_dim=256):
        # Generator (feature extractor)
        self.generator = nn.Sequential(
            nn.Linear(feature_dim, hidden_dim),
            nn.ReLU(),
            nn.Linear(hidden_dim, hidden_dim),
            nn.ReLU(),
            nn.Linear(hidden_dim, feature_dim)
        )

        # Discriminator
        self.discriminator = nn.Sequential(
            nn.Linear(feature_dim, hidden_dim),
            nn.ReLU(),
            nn.Linear(hidden_dim, hidden_dim),
            nn.ReLU(),
            nn.Linear(hidden_dim, 1),
            nn.Sigmoid()
        )

        self.gen_optimizer = torch.optim.Adam(self.generator.parameters(), lr=0.0002)
        self.disc_optimizer = torch.optim.Adam(self.discriminator.parameters(), lr=0.0002)

    def train_adversarial_step(self, sim_features, real_features):
        """
        Adversarial training step
        """
        batch_size = sim_features.size(0)

        # Labels for discriminator
        real_labels = torch.ones(batch_size, 1).to(sim_features.device)
        fake_labels = torch.zeros(batch_size, 1).to(sim_features.device)

        # Train discriminator
        self.disc_optimizer.zero_grad()

        # Real features (should be classified as real)
        disc_real_output = self.discriminator(real_features)
        disc_real_loss = F.binary_cross_entropy(disc_real_output, real_labels)

        # Fake features (simulated, should be classified as fake)
        gen_output = self.generator(sim_features)
        disc_fake_output = self.discriminator(gen_output.detach())
        disc_fake_loss = F.binary_cross_entropy(disc_fake_output, fake_labels)

        disc_loss = disc_real_loss + disc_fake_loss
        disc_loss.backward()
        self.disc_optimizer.step()

        # Train generator (to fool discriminator)
        self.gen_optimizer.zero_grad()
        gen_output = self.generator(sim_features)
        disc_gen_output = self.discriminator(gen_output)
        gen_loss = F.binary_cross_entropy(disc_gen_output, real_labels)  # Want discriminator to think it's real
        gen_loss.backward()
        self.gen_optimizer.step()

        return gen_loss.item(), disc_loss.item()

Self-Supervised Domain Adaptation​

class SelfSupervisedDomainAdapter:
    def __init__(self, encoder_dim=256):
        # Encoder network
        self.encoder = nn.Sequential(
            nn.Linear(encoder_dim, encoder_dim // 2),
            nn.ReLU(),
            nn.Linear(encoder_dim // 2, encoder_dim // 4),
            nn.ReLU(),
            nn.Linear(encoder_dim // 4, 64)  # Latent space
        )

        # Prediction head
        self.predictor = nn.Sequential(
            nn.Linear(64, encoder_dim // 4),
            nn.ReLU(),
            nn.Linear(encoder_dim // 4, encoder_dim // 2)
        )

        self.optimizer = torch.optim.Adam(
            list(self.encoder.parameters()) + list(self.predictor.parameters()),
            lr=0.001
        )

    def contrastive_loss(self, z_i, z_j, temperature=0.5):
        """
        Contrastive loss for self-supervised learning
        """
        batch_size = z_i.size(0)

        # Normalize embeddings
        z_i_norm = F.normalize(z_i, dim=1)
        z_j_norm = F.normalize(z_j, dim=1)

        # Positive similarities
        pos_sim = F.cosine_similarity(z_i_norm, z_j_norm, dim=1).unsqueeze(1)

        # Negative similarities
        neg_sim_ii = torch.mm(z_i_norm, z_i_norm.t())
        neg_sim_ij = torch.mm(z_i_norm, z_j_norm.t())
        neg_sim_ji = torch.mm(z_j_norm, z_i_norm.t())
        neg_sim_jj = torch.mm(z_j_norm, z_j_norm.t())

        # Combine all similarities
        all_sim = torch.cat([
            torch.cat([neg_sim_ii, neg_sim_ij], dim=1),
            torch.cat([neg_sim_ji, neg_sim_jj], dim=1)
        ], dim=0)

        # Remove self-similarities
        mask = torch.eye(2 * batch_size).to(z_i.device)
        all_sim = all_sim.masked_fill(mask.bool(), float('-inf'))

        # Compute loss
        logits = torch.cat([pos_sim, all_sim], dim=1) / temperature
        labels = torch.zeros(2 * batch_size).long().to(z_i.device)

        loss = F.cross_entropy(logits, labels)
        return loss

    def train_self_supervised(self, sim_batch, real_batch):
        """
        Self-supervised training step
        """
        self.optimizer.zero_grad()

        # Encode both domains
        sim_encoded = self.encoder(sim_batch)
        real_encoded = self.encoder(real_batch)

        # Predict representations
        sim_pred = self.predictor(sim_encoded)
        real_pred = self.predictor(real_encoded)

        # Compute contrastive loss
        loss = self.contrastive_loss(sim_pred, real_pred)

        loss.backward()
        self.optimizer.step()

        return loss.item()

Advanced Transfer Techniques​

Sim-to-Real with System Identification​

class SystemIdentificationTransfer:
    def __init__(self):
        self.sim_model_params = {}
        self.real_model_params = {}
        self.mapping_function = None

    def identify_system_parameters(self, input_signal, output_signal):
        """
        Identify system parameters from input-output data
        """
        # For simplicity, assume linear system identification
        # In practice, this would use more sophisticated methods

        # Example: ARX model identification
        # y(k) = -a1*y(k-1) - a2*y(k-2) + b1*u(k-1) + b2*u(k-2)

        n = len(output_signal)
        na, nb = 2, 2  # Model orders

        # Form regression matrix
        phi = np.zeros((n - max(na, nb), na + nb))
        for i in range(max(na, nb), n):
            idx = i - max(na, nb)
            for j in range(na):
                phi[idx, j] = -output_signal[i - j - 1]
            for j in range(nb):
                phi[idx, na + j] = input_signal[i - j - 1]

        # Estimate parameters using least squares
        Y = output_signal[max(na, nb):n]
        theta = np.linalg.lstsq(phi, Y, rcond=None)[0]

        a_est = theta[:na]
        b_est = theta[na:]

        return {'a': a_est, 'b': b_est}

    def learn_mapping_function(self, sim_data, real_data):
        """
        Learn mapping function between simulation and real parameters
        """
        # This could be a neural network, GP, or other function approximator
        from sklearn.gaussian_process import GaussianProcessRegressor
        from sklearn.gaussian_process.kernels import RBF

        # Assume sim_data and real_data are parameter vectors
        gp = GaussianProcessRegressor(kernel=RBF(), n_restarts_optimizer=10)
        gp.fit(sim_data.reshape(-1, 1), real_data)

        self.mapping_function = gp
        return gp

    def adapt_policy(self, sim_policy, adaptation_data=None):
        """
        Adapt policy from simulation to real using system identification
        """
        if self.mapping_function is not None:
            # Apply mapping to policy parameters
            adapted_policy = self.apply_mapping(sim_policy)
        else:
            # Use domain randomization approach
            adapted_policy = self.randomize_and_test(sim_policy)

        return adapted_policy

    def apply_mapping(self, policy_params):
        """
        Apply learned mapping to policy parameters
        """
        if self.mapping_function is not None:
            mapped_params = self.mapping_function.predict(policy_params.reshape(-1, 1))
            return mapped_params.flatten()
        else:
            return policy_params

    def randomize_and_test(self, policy):
        """
        Use domain randomization to find robust policy
        """
        best_policy = policy
        best_performance = float('-inf')

        for _ in range(100):  # Try 100 randomizations
            randomized_params = self.randomize_parameters()
            performance = self.evaluate_policy(policy, randomized_params)

            if performance > best_performance:
                best_performance = performance
                best_policy = policy

        return best_policy

    def randomize_parameters(self):
        """
        Randomize system parameters for testing
        """
        return np.random.uniform(0.5, 1.5, size=10)  # Example

    def evaluate_policy(self, policy, params):
        """
        Evaluate policy with given parameters
        """
        return np.random.random()  # Placeholder

Few-Shot Domain Adaptation​

class FewShotDomainAdapter:
    def __init__(self, meta_learning_rate=0.001):
        self.meta_learner = MetaLearningNetwork()
        self.meta_optimizer = torch.optim.Adam(
            self.meta_learner.parameters(),
            lr=meta_learning_rate
        )

    def meta_train(self, tasks, num_updates=5):
        """
        Meta-training phase
        """
        total_loss = 0

        for task in tasks:
            # Sample support and query sets
            support_set = task['support']
            query_set = task['query']

            # Adapt to specific task
            adapted_params = self.adapt_to_task(
                support_set,
                num_updates=num_updates
            )

            # Evaluate on query set
            query_loss = self.evaluate_on_query(query_set, adapted_params)
            total_loss += query_loss

        # Update meta-learner
        self.meta_optimizer.zero_grad()
        total_loss.backward()
        self.meta_optimizer.step()

        return total_loss.item()

    def adapt_to_task(self, support_set, num_updates=5):
        """
        Adapt meta-learner to specific task using support set
        """
        # Get initial parameters from meta-learner
        params = self.meta_learner.get_params()

        for _ in range(num_updates):
            loss = self.compute_task_loss(support_set, params)
            # Compute gradients and update (inner loop)
            grads = torch.autograd.grad(loss, params.values())

            # Update parameters
            new_params = {}
            for i, (name, param) in enumerate(params.items()):
                new_params[name] = param - 0.01 * grads[i]  # Inner loop LR

            params = new_params

        return params

    def compute_task_loss(self, dataset, params):
        """
        Compute loss for a specific task
        """
        # This would involve forwarding through the network
        # with the given parameters
        return torch.tensor(0.0)  # Placeholder

    def evaluate_on_query(self, query_set, params):
        """
        Evaluate adapted parameters on query set
        """
        return torch.tensor(0.0)  # Placeholder

class MetaLearningNetwork(nn.Module):
    def __init__(self):
        super().__init__()
        self.network = nn.Sequential(
            nn.Linear(784, 256),
            nn.ReLU(),
            nn.Linear(256, 128),
            nn.ReLU(),
            nn.Linear(128, 10)
        )

    def forward(self, x):
        return self.network(x)

    def get_params(self):
        """Return named parameters"""
        return dict(self.named_parameters())

Practical Implementation Strategies​

Progressive Domain Transfer​

class ProgressiveDomainTransfer:
    def __init__(self):
        self.transfer_stages = [
            {
                'name': 'initial_training',
                'method': 'pure_simulation',
                'epochs': 1000,
                'eval_frequency': 10
            },
            {
                'name': 'domain_randomization',
                'method': 'randomization',
                'epochs': 2000,
                'eval_frequency': 5
            },
            {
                'name': 'fine_tuning',
                'method': 'transfer_learning',
                'epochs': 500,
                'eval_frequency': 1
            },
            {
                'name': 'real_world_validation',
                'method': 'online_adaptation',
                'epochs': float('inf'),
                'eval_frequency': 1
            }
        ]

        self.current_stage = 0
        self.stage_progress = 0

    def execute_transfer_stage(self, stage_idx, data_provider):
        """
        Execute a specific transfer stage
        """
        stage = self.transfer_stages[stage_idx]

        if stage['method'] == 'pure_simulation':
            return self.execute_simulation_training(data_provider)
        elif stage['method'] == 'randomization':
            return self.execute_domain_randomization(data_provider)
        elif stage['method'] == 'transfer_learning':
            return self.execute_transfer_learning(data_provider)
        elif stage['method'] == 'online_adaptation':
            return self.execute_online_adaptation(data_provider)

    def execute_simulation_training(self, data_provider):
        """
        Execute pure simulation training
        """
        print("Executing pure simulation training...")
        # Train on simulation data only
        return {'success': True, 'metrics': {'sim_reward': 0.95}}

    def execute_domain_randomization(self, data_provider):
        """
        Execute domain randomization training
        """
        print("Executing domain randomization...")
        # Train with randomized domain parameters
        return {'success': True, 'metrics': {'sim_reward': 0.85, 'variance': 0.1}}

    def execute_transfer_learning(self, data_provider):
        """
        Execute transfer learning with limited real data
        """
        print("Executing transfer learning...")
        # Fine-tune on limited real data
        return {'success': True, 'metrics': {'sim_reward': 0.75, 'real_reward': 0.65}}

    def execute_online_adaptation(self, data_provider):
        """
        Execute online adaptation
        """
        print("Executing online adaptation...")
        # Continuously adapt based on real-world feedback
        return {'success': True, 'metrics': {'real_reward': 0.70}}

Validation and Monitoring​

class TransferValidationSystem:
    def __init__(self):
        self.metrics_history = {
            'sim_performance': [],
            'real_performance': [],
            'transfer_gap': [],
            'domain_similarity': []
        }

    def validate_transfer(self, policy, sim_env, real_env):
        """
        Validate policy transfer across domains
        """
        # Evaluate on simulation
        sim_returns = self.evaluate_policy(policy, sim_env)
        self.metrics_history['sim_performance'].append(sim_returns)

        # Evaluate on real environment
        real_returns = self.evaluate_policy(policy, real_env)
        self.metrics_history['real_performance'].append(real_returns)

        # Calculate transfer gap
        transfer_gap = sim_returns - real_returns
        self.metrics_history['transfer_gap'].append(transfer_gap)

        # Calculate domain similarity
        sim_features = self.extract_features(sim_env)
        real_features = self.extract_features(real_env)
        domain_sim = self.calculate_domain_similarity(sim_features, real_features)
        self.metrics_history['domain_similarity'].append(domain_sim)

        return {
            'sim_return': sim_returns,
            'real_return': real_returns,
            'transfer_gap': transfer_gap,
            'domain_similarity': domain_sim
        }

    def evaluate_policy(self, policy, env):
        """
        Evaluate policy in environment
        """
        total_reward = 0
        state = env.reset()

        for _ in range(100):  # Evaluation horizon
            action = policy.act(state)
            state, reward, done, _ = env.step(action)
            total_reward += reward

            if done:
                break

        return total_reward

    def extract_features(self, env):
        """
        Extract features from environment for comparison
        """
        # This would extract relevant features for domain comparison
        return np.random.rand(10)  # Placeholder

    def calculate_domain_similarity(self, features1, features2):
        """
        Calculate similarity between two feature sets
        """
        # Use some distance measure (e.g., MMD, correlation)
        return np.corrcoef(features1, features2)[0, 1]  # Placeholder

    def generate_transfer_report(self):
        """
        Generate comprehensive transfer validation report
        """
        report = {
            'average_sim_performance': np.mean(self.metrics_history['sim_performance']),
            'average_real_performance': np.mean(self.metrics_history['real_performance']),
            'average_transfer_gap': np.mean(self.metrics_history['transfer_gap']),
            'final_domain_similarity': self.metrics_history['domain_similarity'][-1] if self.metrics_history['domain_similarity'] else 0,
            'transfer_efficiency': self.calculate_transfer_efficiency()
        }

        return report

    def calculate_transfer_efficiency(self):
        """
        Calculate transfer efficiency metric
        """
        if (len(self.metrics_history['sim_performance']) > 0 and
            len(self.metrics_history['real_performance']) > 0):

            avg_sim = np.mean(self.metrics_history['sim_performance'])
            avg_real = np.mean(self.metrics_history['real_performance'])

            if avg_sim > 0:
                return avg_real / avg_sim
            else:
                return 0
        return 0

Best Practices and Guidelines​

1. Gradual Complexity Increase​

  • Start with simple environments and gradually increase complexity
  • Use curriculum learning approaches
  • Validate at each complexity level

2. Systematic Randomization​

  • Randomize parameters that are known to differ between sim and real
  • Use wide ranges to ensure robustness
  • Monitor performance during randomization

3. Real-World Feedback Loop​

  • Collect real-world data to inform simulation improvements
  • Use Bayesian optimization for hyperparameter tuning
  • Implement online adaptation mechanisms

4. Validation Strategy​

  • Use multiple validation metrics
  • Test across diverse conditions
  • Monitor long-term performance stability

Learning Objectives​

By the end of this lesson, you should be able to:

  • Understand the sim-to-real transfer problem and its challenges
  • Implement domain randomization techniques to improve policy robustness
  • Apply domain adaptation methods to bridge simulation-reality gaps
  • Use advanced techniques like system identification and meta-learning for transfer
  • Validate and monitor sim-to-real transfer performance
  • Apply best practices for progressive domain transfer
  • Design comprehensive validation strategies for transfer learning
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