Geometric regularization of autoencoders via observed stochastic dynamics

1. 1
   Understand System Dynamics

   Analyze your high-dimensional time-series data from a stochastic dynamical system, identifying characteristics like long-timescale behavior and potential low-dimensional manifolds where the system evolves.
2. 2
   Select Autoencoder Base

   Choose an appropriate autoencoder architecture (e.g., Variational Autoencoder, Denoising Autoencoder, or a custom design) as your base model for dimensionality reduction.
3. 3
   Define Geometric Regularization

   Research and select a geometric regularization technique suitable for your data's intrinsic manifold structure. This might involve preserving local distances, tangent spaces, geodesic paths, or other topological properties in the latent space.
4. 4
   Implement Custom Loss Function

   Add a custom loss component to your autoencoder's objective function. This component will quantify how well the latent space preserves the desired geometric properties of the input data, acting as the geometric regularization term.
5. 5
   Train and Evaluate

   Train the autoencoder with the combined reconstruction loss and your custom geometric regularization loss. Evaluate its ability to learn robust low-dimensional representations and accurately capture long-term system behavior from the latent space.
6. 6
   Construct Reduced Simulator

   Utilize the trained autoencoder's encoder and decoder components to form a reduced-order simulator. This simulator will operate on the learned low-dimensional latent space to model and predict the system's long-timescale dynamics.

import tensorflow as tf
from tensorflow.keras import layers, models, losses
import numpy as np

# Placeholder for a complex geometric loss function.
# In a real scenario, this would compute a metric
# like preserving local distances, tangent spaces, or specific manifold properties.
def geometric_loss_function(original_data, latent_representation):
    # Example: a simple L2 regularization on the latent space for illustration.
    # A true geometric loss would compare structures between original_data and latent_representation.
    return tf.reduce_mean(tf.square(latent_representation)) * 0.01

class GeoRegularizedAutoencoder(models.Model):
    def __init__(self, original_dim, latent_dim):
        super(GeoRegularizedAutoencoder, self).__init__()
        self.encoder = models.Sequential([
            layers.InputLayer(input_shape=(original_dim,)),
            layers.Dense(128, activation='relu'),
            layers.Dense(64, activation='relu'),
            layers.Dense(latent_dim, activation='relu')
        ])
        self.decoder = models.Sequential([
            layers.InputLayer(input_shape=(latent_dim,)),
            layers.Dense(64, activation='relu'),
            layers.Dense(128, activation='relu'),
            layers.Dense(original_dim, activation='linear') # Use 'linear' for continuous, unbounded data
        ])

    def call(self, x):
        latent = self.encoder(x)
        decoded = self.decoder(latent)
        return decoded, latent

    def train_step(self, data):
        with tf.GradientTape() as tape:
            decoded, latent = self(data)
            # Reconstruction loss (e.g., Mean Squared Error)
            reconstruction_loss = losses.MeanSquaredError()(data, decoded)
            # Geometric regularization loss
            geo_loss = geometric_loss_function(data, latent)
            
            # Combine losses. The weight of geo_loss is a hyperparameter.
            total_loss = reconstruction_loss + geo_loss

        grads = tape.gradient(total_loss, self.trainable_weights)
        self.optimizer.apply_gradients(zip(grads, self.trainable_weights))
        
        return {'loss': total_loss, 'reconstruction_loss': reconstruction_loss, 'geometric_loss': geo_loss}

# --- Example Usage --- 
original_dim = 100 # Dimension of your input data (e.g., features per time step)
latent_dim = 10    # Desired dimension of the low-dimensional manifold

# 1. Instantiate the autoencoder
autoencoder = GeoRegularizedAutoencoder(original_dim, latent_dim)

# 2. Compile the model
autoencoder.compile(optimizer=tf.keras.optimizers.Adam(learning_rate=0.001))

# 3. Generate dummy data (replace with your actual high-dimensional time-series data)
x_train = np.random.rand(1000, original_dim).astype('float32') # 1000 samples, 100 features

# 4. Train the autoencoder
print("\nStarting training...")
history = autoencoder.fit(x_train, epochs=5, batch_size=32)
print("Training complete.")

# 5. Use the trained model
# Get latent representations
_, latent_representations = autoencoder(x_train)
print(f"\nShape of latent representations: {latent_representations.shape}")

# Reconstruct data from latent space
reconstructed_data, _ = autoencoder(x_train)
print(f"Shape of reconstructed data: {reconstructed_data.shape}")

• Paperarxiv.org