Tejas-Anvekar/Viscous_Cahn_Hilliard_2D_Spatio-Temporal

Dataset Card: Viscous Cahn-Hilliard Optimal Control

Dataset Summary

This dataset contains 2,000 high-fidelity simulations of the Viscous Cahn-Hilliard (vCH) equation under randomized control forcing. It was generated to support research into Sparse Optimal Control, SciML (Scientific Machine Learning), and Phase Field Modeling. Official Code Repository: Sparse-optimal-control-of-Viscous-Chan-hilliard (GitHub) Each sample represents the evolution of a two-phase system (separation of immiscible components) steered by an external control field. The data was generated using a custom high-precision Finite Difference solver with Crank-Nicolson time integration.

Dataset Structure

Tensor Shape

The data is stored as a 5D tensor (logically), though physically sharded into Parquet files.

• —Shape: (2000, 101, 128, 128, 4)
• —Dimensions: (Samples, Time Steps, Height, Width, Channels)

Channel Definitions

Based on the data_process.py pipeline, the 4 channels correspond to:

1. 1.Channel 0 (`phi` / $\varphi$): The phase-field order parameter ($\varphi \in [-1, 1]$). Represents the concentration difference between the two phases.
2. 2.Channel 1 (`mu` / $\mu$): The chemical potential. Defined as $\mu = -\kappa\Delta\varphi + f'(\varphi) - w$.
3. 3.Channel 2 (`w`): The auxiliary control variable. Governed by the relaxation equation $\gamma \partial_t w + w = u$.
4. 4.Channel 3 (`u`): The external control forcing field applied to the system.

Physical Domain

• —Spatial Domain: $\Omega = [0, 1] \times [0, 1]$
• —Grid Resolution: $128 \times 128$ (uniform grid spacing $h \approx 0.0078$).
• —Time Horizon: $T = 1.0$ seconds.
• —Time Resolution: $dt = 0.01$ (101 frames per simulation).

Data Generation Details

1. The Physics Model

The data follows the Viscous Cahn-Hilliard System with a logarithmic potential, ensuring the phase field stays strictly within physical bounds $(-1, 1)$.

$$ \begin{aligned} \partialt \varphi - \Delta \mu &= 0 \\ \tau \partialt \varphi - \kappa \Delta \varphi + f'(\varphi) &= \mu + w \\ \gamma \partial_t w + w &= u \end{aligned} $$

2. Numerical Implementation

The solver (Forward2_solver.py) utilizes:

• —Spatial Discretization: Standard 5-point Finite Difference Stencil with Neumann Boundary Conditions (Ghost Points).
• —Time Integration: Semi-Implicit Crank-Nicolson scheme.
• —Implicit: Linear diffusion and convex part of the potential ($c_1$).
• —Explicit: Concave part of the potential ($c_2$).
• —Nonlinear Solver: Monolithic Newton-Raphson with Armijo backtracking line search.
• —Mass Conservation: Enforced via a global Lagrange multiplier correction step.

3. Simulation Parameters

The dataset was generated using the configuration from config.py:

4. Input Control Generation

The control fields u were generated procedurally using Smooth Random Fields:

• —Random noise is generated in Fourier space.
• —A low-pass filter ($K^{-1.5}$) is applied to ensure smoothness.
• —Keyframes are interpolated over time using cubic splines to create continuous, varying forces.

How to Load (Python)

import pandas as pd
import numpy as np
from datasets import load_dataset

# 1. Load the dataset (streaming recommended for 50GB)
dataset = load_dataset("your-username/your-repo-name", split="train", streaming=True)

# 2. Iterate through samples
for row in dataset:
    # 3. Reshape the flattened arrays back to (128, 128, 4)
    # Assuming your Parquet conversion flattened the spatial dims
    sample_id = row['sample_id']
    timestep = row['time_step']
    
    # Shape depends on your specific parquet conversion script
    # Example for flattened 1D array column:
    data_flat = np.array(row['readings']) 
    frame_tensor = data_flat.reshape(128, 128, 4)
    
    phi = frame_tensor[:, :, 0] # Phase field
    u   = frame_tensor[:, :, 3] # Control input
    
    print(f"Sample {sample_id} at t={timestep}: Phi range [{phi.min():.2f}, {phi.max():.2f}]")
    break