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notebook-69bdc636-ef92-4d94-abcf-d498ae44584e-paper-20260311-183050.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# An AI Super-Resolution Field Emulator for Cosmological Hydrodynamics: The Lyman-α Forest\n",
"\n",
"**Paper Authors:** Fatemeh Hafezianzadeh, Xiaowen Zhang, Yueying Ni, Rupert A. C. Croft, Tiziana DiMatteo, Mahdi Qezlou, Simeon Bird\n",
"\n",
"## Overview\n",
"\n",
"This notebook provides an educational implementation of the computational workflows described in the paper. The paper presents a **two-stage deep learning framework** to emulate high-resolution cosmological hydrodynamic simulations for modeling the Lyman-α forest.\n",
"\n",
"### Key Innovation\n",
"\n",
"The framework achieves a **~450× speedup** over full smoothed particle hydrodynamics (SPH) simulations while maintaining high accuracy:\n",
"- Subpercent error for density, temperature, velocity, and optical depth fields\n",
"- 1.07% mean relative error in flux power spectrum\n",
"- <10% error in flux probability distribution function\n",
"\n",
"### Two-Stage Architecture\n",
"\n",
"1. **Stage 1 - HydroSR**: Stochastic super-resolution GAN that generates high-resolution baryonic fields from low-resolution hydrodynamic simulations\n",
"2. **Stage 2 - HydroEmu**: Deterministic emulator that refines HydroSR outputs using high-resolution initial conditions\n",
"\n",
"### Important Notes on This Notebook\n",
"\n",
"**This is an educational demonstration notebook** designed to run within resource constraints (4GB RAM, <10 minutes execution time). It:\n",
"- Uses **small-scale synthetic data** for demonstration\n",
"- Shows **architecture implementations** without full training (which requires GPUs and hours/days)\n",
"- Implements **key validation metrics** and analysis workflows\n",
"- Provides **clear guidance** on scaling to full production use\n",
"\n",
"For production use, you would need:\n",
"- GPU infrastructure (A100 or similar)\n",
"- Full MP-Gadget simulation data (~20 paired LR/HR simulation boxes)\n",
"- Multiple days of training time\n",
"- Significant storage for simulation outputs"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 1. Setup and Dependencies"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# Install required dependencies\n",
"!uv pip install numpy scipy matplotlib torch torchvision scikit-learn"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"from scipy import stats\n",
"from scipy.optimize import curve_fit\n",
"import torch\n",
"import torch.nn as nn\n",
"import torch.nn.functional as F\n",
"from sklearn.metrics import mean_squared_error\n",
"import warnings\n",
"warnings.filterwarnings('ignore')\n",
"\n",
"# Set random seeds for reproducibility\n",
"np.random.seed(42)\n",
"torch.manual_seed(42)\n",
"if torch.cuda.is_available():\n",
" torch.cuda.manual_seed(42)\n",
"\n",
"print(\"PyTorch version:\", torch.__version__)\n",
"print(\"CUDA available:\", torch.cuda.is_available())\n",
"print(\"Device:\", \"cuda\" if torch.cuda.is_available() else \"cpu\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 2. Synthetic Data Generation\n",
"\n",
"Since running full MP-Gadget simulations is beyond our resource constraints, we generate **synthetic training data** that mimics the structure of real cosmological hydrodynamic simulations.\n",
"\n",
"### Data Characteristics\n",
"\n",
"According to the paper:\n",
"- **Low-resolution (LR)**: 64³ particles in 50 h⁻¹ Mpc box\n",
"- **High-resolution (HR)**: 512³ particles in 50 h⁻¹ Mpc box \n",
"- **Sightlines**: 3600 regularly spaced, 540 pixels each at 10 km/s resolution\n",
"- **Fields**: 8 channels (3 displacement, 3 velocity, 1 internal energy, 1 gas/star label)\n",
"\n",
"For this demonstration, we use much smaller grids (16³ → 32³) to stay within memory limits."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"class SyntheticCosmologyData:\n",
" \"\"\"\n",
" Generate synthetic cosmological simulation data for demonstration.\n",
" \n",
" In production, this would be replaced by actual MP-Gadget simulation outputs.\n",
" \"\"\"\n",
" def __init__(self, lr_resolution=16, hr_resolution=32, box_size=50.0, n_samples=20):\n",
" \"\"\"\n",
" Args:\n",
" lr_resolution: Grid resolution for low-resolution simulation (paper uses 64)\n",
" hr_resolution: Grid resolution for high-resolution simulation (paper uses 512)\n",
" box_size: Simulation box size in h^-1 Mpc (paper uses 50)\n",
" n_samples: Number of paired simulation examples (paper uses 20 total, 16 train)\n",
" \"\"\"\n",
" self.lr_res = lr_resolution\n",
" self.hr_res = hr_resolution\n",
" self.box_size = box_size\n",
" self.n_samples = n_samples\n",
" self.scale_factor = hr_resolution // lr_resolution\n",
" \n",
" def generate_density_field(self, resolution, power_index=-2.5):\n",
" \"\"\"\n",
" Generate a realistic density field using power-law spectrum.\n",
" Mimics cosmic web structure.\n",
" \"\"\"\n",
" # Generate Gaussian random field in Fourier space\n",
" shape = (resolution, resolution, resolution)\n",
" k_field = np.fft.fftn(np.random.randn(*shape))\n",
" \n",
" # Apply power-law filter (P(k) ~ k^power_index)\n",
" kx = np.fft.fftfreq(resolution) * resolution\n",
" ky = np.fft.fftfreq(resolution) * resolution\n",
" kz = np.fft.fftfreq(resolution) * resolution\n",
" KX, KY, KZ = np.meshgrid(kx, ky, kz, indexing='ij')\n",
" K = np.sqrt(KX**2 + KY**2 + KZ**2)\n",
" K[K == 0] = 1.0 # Avoid division by zero\n",
" \n",
" # Apply power spectrum filter\n",
" power_filter = K**(power_index/2.0)\n",
" filtered_k = k_field * power_filter\n",
" \n",
" # Transform back to real space\n",
" density = np.real(np.fft.ifftn(filtered_k))\n",
" \n",
" # Convert to overdensity (ρ/ρ̄ - 1) and shift to ensure positivity\n",
" density = (density - density.mean()) / density.std()\n",
" density = np.exp(density * 0.5) # Log-normal transformation\n",
" \n",
" return density\n",
" \n",
" def generate_velocity_field(self, density_field):\n",
" \"\"\"\n",
" Generate velocity field correlated with density (Zel'dovich approximation).\n",
" \"\"\"\n",
" # Velocity is related to density gradient in linear regime\n",
" grad = np.gradient(density_field)\n",
" vx = grad[0] * 100.0 # Scale to ~100 km/s\n",
" vy = grad[1] * 100.0\n",
" vz = grad[2] * 100.0\n",
" return vx, vy, vz\n",
" \n",
" def generate_temperature_field(self, density_field, T0=1.6e4, gamma=1.44):\n",
" \"\"\"\n",
" Generate temperature field using temperature-density relation.\n",
" T = T0 * (ρ/ρ̄)^(γ-1) with scatter\n",
" \"\"\"\n",
" # Power-law relation with log-normal scatter\n",
" temperature = T0 * density_field**(gamma - 1)\n",
" # Add scatter (NRMSE ~ 0.05% from paper)\n",
" scatter = np.random.lognormal(0, 0.15, density_field.shape)\n",
" temperature *= scatter\n",
" return temperature\n",
" \n",
" def generate_internal_energy(self, temperature):\n",
" \"\"\"\n",
" Convert temperature to internal energy.\n",
" For ideal gas: u = (3/2) * k_B * T / (μ * m_p)\n",
" \"\"\"\n",
" # Simplified conversion (actual units don't matter for demo)\n",
" return temperature * 1e-4\n",
" \n",
" def generate_gas_star_labels(self, density_field, threshold=1000):\n",
" \"\"\"\n",
" Generate gas/star classification labels.\n",
" Paper uses quick-Lyα approximation: gas -> star when ρ/ρ̄ > 1000\n",
" \"\"\"\n",
" return (density_field < threshold).astype(np.float32)\n",
" \n",
" def generate_sample_pair(self):\n",
" \"\"\"\n",
" Generate one paired LR-HR simulation sample.\n",
" \n",
" Returns:\n",
" lr_data: Low-resolution 8-channel field (C, D, H, W)\n",
" hr_data: High-resolution 8-channel field (C, D, H, W)\n",
" \"\"\"\n",
" # Generate HR simulation (ground truth)\n",
" hr_density = self.generate_density_field(self.hr_res)\n",
" hr_vx, hr_vy, hr_vz = self.generate_velocity_field(hr_density)\n",
" hr_temperature = self.generate_temperature_field(hr_density)\n",
" hr_energy = self.generate_internal_energy(hr_temperature)\n",
" hr_gas_label = self.generate_gas_star_labels(hr_density)\n",
" \n",
" # HR displacement (simplified - in reality from initial conditions)\n",
" hr_dx = np.random.randn(self.hr_res, self.hr_res, self.hr_res) * 0.1\n",
" hr_dy = np.random.randn(self.hr_res, self.hr_res, self.hr_res) * 0.1\n",
" hr_dz = np.random.randn(self.hr_res, self.hr_res, self.hr_res) * 0.1\n",
" \n",
" # Generate LR simulation (coarse resolution)\n",
" lr_density = self.generate_density_field(self.lr_res)\n",
" lr_vx, lr_vy, lr_vz = self.generate_velocity_field(lr_density)\n",
" lr_temperature = self.generate_temperature_field(lr_density)\n",
" lr_energy = self.generate_internal_energy(lr_temperature)\n",
" lr_gas_label = self.generate_gas_star_labels(lr_density)\n",
" \n",
" # LR displacement\n",
" lr_dx = np.random.randn(self.lr_res, self.lr_res, self.lr_res) * 0.1\n",
" lr_dy = np.random.randn(self.lr_res, self.lr_res, self.lr_res) * 0.1\n",
" lr_dz = np.random.randn(self.lr_res, self.lr_res, self.lr_res) * 0.1\n",
" \n",
" # Stack into 8-channel format: [dx, dy, dz, vx, vy, vz, energy, gas_label]\n",
" lr_data = np.stack([lr_dx, lr_dy, lr_dz, lr_vx, lr_vy, lr_vz, lr_energy, lr_gas_label], axis=0)\n",
" hr_data = np.stack([hr_dx, hr_dy, hr_dz, hr_vx, hr_vy, hr_vz, hr_energy, hr_gas_label], axis=0)\n",
" \n",
" # Also return auxiliary fields for validation\n",
" aux_data = {\n",
" 'hr_density': hr_density,\n",
" 'hr_temperature': hr_temperature,\n",
" 'lr_density': lr_density,\n",
" 'lr_temperature': lr_temperature\n",
" }\n",
" \n",
" return lr_data.astype(np.float32), hr_data.astype(np.float32), aux_data\n",
" \n",
" def generate_dataset(self, train_split=0.8):\n",
" \"\"\"\n",
" Generate full dataset of paired simulations.\n",
" \n",
" Returns:\n",
" train_lr, train_hr, test_lr, test_hr: Training and test sets\n",
" \"\"\"\n",
" n_train = int(self.n_samples * train_split)\n",
" \n",
" train_lr, train_hr = [], []\n",
" test_lr, test_hr = [], []\n",
" aux_train, aux_test = [], []\n",
" \n",
" print(f\"Generating {self.n_samples} synthetic simulation pairs...\")\n",
" for i in range(self.n_samples):\n",
" lr, hr, aux = self.generate_sample_pair()\n",
" if i < n_train:\n",
" train_lr.append(lr)\n",
" train_hr.append(hr)\n",
" aux_train.append(aux)\n",
" else:\n",
" test_lr.append(lr)\n",
" test_hr.append(hr)\n",
" aux_test.append(aux)\n",
" \n",
" print(f\"Generated {n_train} training and {self.n_samples - n_train} test samples\")\n",
" \n",
" return (\n",
" np.array(train_lr), np.array(train_hr),\n",
" np.array(test_lr), np.array(test_hr),\n",
" aux_train, aux_test\n",
" )\n",
"\n",
"# Generate synthetic dataset\n",
"data_generator = SyntheticCosmologyData(lr_resolution=16, hr_resolution=32, n_samples=10)\n",
"train_lr, train_hr, test_lr, test_hr, aux_train, aux_test = data_generator.generate_dataset()\n",
"\n",
"print(f\"\\nDataset shapes:\")\n",
"print(f\" Training LR: {train_lr.shape} (samples, channels, depth, height, width)\")\n",
"print(f\" Training HR: {train_hr.shape}\")\n",
"print(f\" Test LR: {test_lr.shape}\")\n",
"print(f\" Test HR: {test_hr.shape}\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 3. Stage 1: HydroSR - Stochastic Super-Resolution Model\n",
"\n",
"The **HydroSR** model is a GAN-based super-resolution network that transforms low-resolution inputs into high-resolution outputs. It uses:\n",
"\n",
"- **Generator**: Hierarchical architecture with multi-scale processing (adapted from Ni et al. 2021)\n",
"- **Discriminator**: PatchGAN with residual connections\n",
"- **Loss**: Combination of supervised MSE (Lagrangian + Eulerian) + WGAN-GP adversarial loss\n",
"\n",
"### Architecture Details\n",
"\n",
"From the paper (Section 2.2, Equation 1):\n",
"$$L_{\\text{total}} = L_{\\text{Lag}}^{\\text{MSE}} + L_{\\text{Eul}}^{\\text{MSE}} + \\lambda_{\\text{adv}} L_{\\text{adv}}^{\\text{WGAN-GP}}$$\n",
"\n",
"Where WGAN-GP loss (Equation 3) is:\n",
"$$L^{\\text{WGAN-GP}} = \\mathbb{E}_{\\ell,z}[D(\\ell, G(\\ell,z))] - \\mathbb{E}_{\\ell,h}[D(\\ell,h)] + \\lambda \\mathbb{E}_{\\ell,\\hat{h}}[(\\|\\nabla_\\hat{i} D(\\ell,\\hat{i})\\|_2 - 1)^2]$$"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"class HydroSRGenerator(nn.Module):\n",
" \"\"\"\n",
" Simplified HydroSR Generator architecture.\n",
" \n",
" Based on the hierarchical structure from Ni et al. (2021).\n",
" In production, this would be much deeper with multiple resolution levels.\n",
" \"\"\"\n",
" def __init__(self, in_channels=8, out_channels=8, base_filters=32, scale_factor=2):\n",
" super(HydroSRGenerator, self).__init__()\n",
" self.scale_factor = scale_factor\n",
" \n",
" # Encoder path (lower branch in paper's Figure 1)\n",
" self.enc1 = nn.Sequential(\n",
" nn.Conv3d(in_channels, base_filters, 3, padding=1),\n",
" nn.LeakyReLU(0.2)\n",
" )\n",
" self.enc2 = nn.Sequential(\n",
" nn.Conv3d(base_filters, base_filters*2, 3, padding=1),\n",
" nn.LeakyReLU(0.2)\n",
" )\n",
" self.enc3 = nn.Sequential(\n",
" nn.Conv3d(base_filters*2, base_filters*4, 3, padding=1),\n",
" nn.LeakyReLU(0.2)\n",
" )\n",
" \n",
" # Projection layers (upper branch in paper's Figure 1)\n",
" self.proj1 = nn.Conv3d(base_filters, out_channels, 1)\n",
" self.proj2 = nn.Conv3d(base_filters*2, out_channels, 1)\n",
" self.proj3 = nn.Conv3d(base_filters*4, out_channels, 1)\n",
" \n",
" def forward(self, x):\n",
" # Multi-scale processing\n",
" feat1 = self.enc1(x)\n",
" feat2 = self.enc2(feat1)\n",
" feat3 = self.enc3(feat2)\n",
" \n",
" # Project and upsample at each scale\n",
" out1 = self.proj1(feat1)\n",
" out2 = self.proj2(feat2)\n",
" out3 = self.proj3(feat3)\n",
" \n",
" # Trilinear interpolation to target resolution\n",
" target_size = (x.shape[2] * self.scale_factor,\n",
" x.shape[3] * self.scale_factor,\n",
" x.shape[4] * self.scale_factor)\n",
" \n",
" out1_up = F.interpolate(out1, size=target_size, mode='trilinear', align_corners=False)\n",
" out2_up = F.interpolate(out2, size=target_size, mode='trilinear', align_corners=False)\n",
" out3_up = F.interpolate(out3, size=target_size, mode='trilinear', align_corners=False)\n",
" \n",
" # Accumulate outputs across levels\n",
" output = out1_up + out2_up + out3_up\n",
" \n",
" return output\n",
"\n",
"\n",
"class PatchGANDiscriminator(nn.Module):\n",
" \"\"\"\n",
" PatchGAN discriminator with residual connections.\n",
" \n",
" Evaluates local patches and estimates Wasserstein distance.\n",
" \"\"\"\n",
" def __init__(self, in_channels=8, base_filters=32):\n",
" super(PatchGANDiscriminator, self).__init__()\n",
" \n",
" self.model = nn.Sequential(\n",
" # Layer 1\n",
" nn.Conv3d(in_channels, base_filters, 4, stride=2, padding=1),\n",
" nn.LeakyReLU(0.2),\n",
" \n",
" # Layer 2\n",
" nn.Conv3d(base_filters, base_filters*2, 4, stride=2, padding=1),\n",
" nn.LeakyReLU(0.2),\n",
" \n",
" # Layer 3\n",
" nn.Conv3d(base_filters*2, base_filters*4, 4, stride=2, padding=1),\n",
" nn.LeakyReLU(0.2),\n",
" \n",
" # Output layer - single channel output (Wasserstein distance)\n",
" nn.Conv3d(base_filters*4, 1, 4, stride=1, padding=1)\n",
" )\n",
" \n",
" def forward(self, x):\n",
" return self.model(x)\n",
"\n",
"\n",
"# Instantiate models\n",
"device = torch.device('cuda' if torch.cuda.is_available() else 'cpu')\n",
"hydrosr_gen = HydroSRGenerator(in_channels=8, out_channels=8, scale_factor=2).to(device)\n",
"hydrosr_disc = PatchGANDiscriminator(in_channels=8).to(device)\n",
"\n",
"print(\"HydroSR Generator:\")\n",
"print(f\" Parameters: {sum(p.numel() for p in hydrosr_gen.parameters()):,}\")\n",
"print(f\"\\nHydroSR Discriminator:\")\n",
"print(f\" Parameters: {sum(p.numel() for p in hydrosr_disc.parameters()):,}\")\n",
"\n",
"# Test forward pass\n",
"test_input = torch.randn(1, 8, 16, 16, 16).to(device)\n",
"test_output = hydrosr_gen(test_input)\n",
"print(f\"\\nTest: Input shape {test_input.shape} -> Output shape {test_output.shape}\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 4. Stage 2: HydroEmu - Deterministic Emulator\n",
"\n",
"The **HydroEmu** model refines the HydroSR output using high-resolution initial conditions. It uses a **U-Net architecture** with:\n",
"\n",
"- **Input**: 16 channels (8 from HydroSR + 8 from HR initial conditions)\n",
"- **Architecture**: Residual blocks with group normalization and SiLU activation\n",
"- **Training**: Same loss as HydroSR but Eulerian loss only on density field\n",
"\n",
"From the paper (Section 2.2):\n",
"> \"The input to the network is constructed by concatenating the 8-channel output of the HydroSR model with 8 additional channels derived from the HR-HydroICs.\""
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"class ResidualBlock(nn.Module):\n",
" \"\"\"\n",
" Residual block with group normalization and SiLU activation.\n",
" \"\"\"\n",
" def __init__(self, channels, num_groups=8):\n",
" super(ResidualBlock, self).__init__()\n",
" self.conv1 = nn.Conv3d(channels, channels, 3, padding=1)\n",
" self.gn1 = nn.GroupNorm(num_groups, channels)\n",
" self.conv2 = nn.Conv3d(channels, channels, 3, padding=1)\n",
" self.gn2 = nn.GroupNorm(num_groups, channels)\n",
" self.silu = nn.SiLU()\n",
" \n",
" def forward(self, x):\n",
" residual = x\n",
" out = self.silu(self.gn1(self.conv1(x)))\n",
" out = self.gn2(self.conv2(out))\n",
" out = self.silu(out + residual)\n",
" return out\n",
"\n",
"\n",
"class HydroEmuGenerator(nn.Module):\n",
" \"\"\"\n",
" HydroEmu U-Net architecture with residual blocks.\n",
" \n",
" Based on Zhang et al. (2025) architecture.\n",
" \"\"\"\n",
" def __init__(self, in_channels=16, out_channels=8, base_filters=32):\n",
" super(HydroEmuGenerator, self).__init__()\n",
" \n",
" # Encoder (downsampling path)\n",
" self.enc1 = nn.Sequential(\n",
" nn.Conv3d(in_channels, base_filters, 3, padding=1),\n",
" ResidualBlock(base_filters)\n",
" )\n",
" self.down1 = nn.Conv3d(base_filters, base_filters*2, 3, stride=2, padding=1)\n",
" \n",
" self.enc2 = nn.Sequential(\n",
" ResidualBlock(base_filters*2),\n",
" ResidualBlock(base_filters*2)\n",
" )\n",
" self.down2 = nn.Conv3d(base_filters*2, base_filters*4, 3, stride=2, padding=1)\n",
" \n",
" # Bottleneck\n",
" self.bottleneck = nn.Sequential(\n",
" ResidualBlock(base_filters*4),\n",
" ResidualBlock(base_filters*4)\n",
" )\n",
" \n",
" # Decoder (upsampling path)\n",
" self.up2 = nn.ConvTranspose3d(base_filters*4, base_filters*2, 3, stride=2, padding=1, output_padding=1)\n",
" self.dec2 = nn.Sequential(\n",
" ResidualBlock(base_filters*4), # Concatenated with skip connection\n",
" ResidualBlock(base_filters*4),\n",
" nn.Conv3d(base_filters*4, base_filters*2, 1)\n",
" )\n",
" \n",
" self.up1 = nn.ConvTranspose3d(base_filters*2, base_filters, 3, stride=2, padding=1, output_padding=1)\n",
" self.dec1 = nn.Sequential(\n",
" ResidualBlock(base_filters*2), # Concatenated with skip connection\n",
" ResidualBlock(base_filters*2),\n",
" nn.Conv3d(base_filters*2, base_filters, 1)\n",
" )\n",
" \n",
" # Output layer\n",
" self.output = nn.Conv3d(base_filters, out_channels, 1)\n",
" \n",
" def forward(self, x):\n",
" # Encoder with skip connections\n",
" enc1 = self.enc1(x)\n",
" down1 = self.down1(enc1)\n",
" \n",
" enc2 = self.enc2(down1)\n",
" down2 = self.down2(enc2)\n",
" \n",
" # Bottleneck\n",
" bottleneck = self.bottleneck(down2)\n",
" \n",
" # Decoder with skip connections\n",
" up2 = self.up2(bottleneck)\n",
" dec2 = self.dec2(torch.cat([up2, enc2], dim=1))\n",
" \n",
" up1 = self.up1(dec2)\n",
" dec1 = self.dec1(torch.cat([up1, enc1], dim=1))\n",
" \n",
" output = self.output(dec1)\n",
" return output\n",
"\n",
"\n",
"# Instantiate HydroEmu\n",
"hydroemu_gen = HydroEmuGenerator(in_channels=16, out_channels=8).to(device)\n",
"\n",
"print(\"HydroEmu Generator (U-Net):\")\n",
"print(f\" Parameters: {sum(p.numel() for p in hydroemu_gen.parameters()):,}\")\n",
"\n",
"# Test forward pass\n",
"test_input_emu = torch.randn(1, 16, 32, 32, 32).to(device) # 16 channels: 8 from HydroSR + 8 from HR-IC\n",
"test_output_emu = hydroemu_gen(test_input_emu)\n",
"print(f\"\\nTest: Input shape {test_input_emu.shape} -> Output shape {test_output_emu.shape}\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 5. Training Demonstration (Conceptual)\n",
"\n",
"**IMPORTANT**: Full training requires:\n",
"- GPU infrastructure (paper used A100 GPUs)\n",
"- Multiple days of training time\n",
"- Large batch processing and data loading\n",
"\n",
"This section shows the **training loop structure** without actually executing it (which would exceed our resource constraints).\n",
"\n",
"### Loss Functions\n",
"\n",
"From the paper, the total loss combines:\n",
"1. **Lagrangian MSE**: Pixel-wise error on particle fields\n",
"2. **Eulerian MSE**: Error after cloud-in-cell (CIC) deposition to grid\n",
"3. **WGAN-GP adversarial loss**: Encourages realistic outputs"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def compute_mse_loss(pred, target):\n",
" \"\"\"\n",
" Compute MSE loss (Equation 2 in paper).\n",
" \"\"\"\n",
" return F.mse_loss(pred, target)\n",
"\n",
"\n",
"def compute_gradient_penalty(discriminator, real_data, fake_data, device):\n",
" \"\"\"\n",
" Compute gradient penalty for WGAN-GP (part of Equation 3).\n",
" \"\"\"\n",
" batch_size = real_data.shape[0]\n",
" alpha = torch.rand(batch_size, 1, 1, 1, 1).to(device)\n",
" interpolates = (alpha * real_data + (1 - alpha) * fake_data).requires_grad_(True)\n",
" \n",
" d_interpolates = discriminator(interpolates)\n",
" \n",
" gradients = torch.autograd.grad(\n",
" outputs=d_interpolates,\n",
" inputs=interpolates,\n",
" grad_outputs=torch.ones_like(d_interpolates),\n",
" create_graph=True,\n",
" retain_graph=True\n",
" )[0]\n",
" \n",
" gradients = gradients.view(batch_size, -1)\n",
" gradient_penalty = ((gradients.norm(2, dim=1) - 1) ** 2).mean()\n",
" \n",
" return gradient_penalty\n",
"\n",
"\n",
"def train_hydrosr_one_epoch(generator, discriminator, train_loader, \n",
" g_optimizer, d_optimizer, device, lambda_adv=1.0, lambda_gp=10.0):\n",
" \"\"\"\n",
" Training loop for one epoch of HydroSR.\n",
" \n",
" In production, this would be called for many epochs with checkpointing,\n",
" learning rate scheduling, and validation monitoring.\n",
" \"\"\"\n",
" generator.train()\n",
" discriminator.train()\n",
" \n",
" for batch_idx, (lr_data, hr_data) in enumerate(train_loader):\n",
" lr_data = lr_data.to(device)\n",
" hr_data = hr_data.to(device)\n",
" \n",
" # Train Discriminator\n",
" d_optimizer.zero_grad()\n",
" \n",
" fake_data = generator(lr_data)\n",
" \n",
" d_real = discriminator(hr_data)\n",
" d_fake = discriminator(fake_data.detach())\n",
" \n",
" # Wasserstein loss\n",
" d_loss_real = -d_real.mean()\n",
" d_loss_fake = d_fake.mean()\n",
" \n",
" # Gradient penalty\n",
" gp = compute_gradient_penalty(discriminator, hr_data, fake_data.detach(), device)\n",
" \n",
" d_loss = d_loss_real + d_loss_fake + lambda_gp * gp\n",
" d_loss.backward()\n",
" d_optimizer.step()\n",
" \n",
" # Train Generator\n",
" g_optimizer.zero_grad()\n",
" \n",
" fake_data = generator(lr_data)\n",
" \n",
" # Lagrangian MSE loss\n",
" mse_lagrangian = compute_mse_loss(fake_data, hr_data)\n",
" \n",
" # Eulerian MSE loss (simplified - in production, apply CIC deposition)\n",
" mse_eulerian = compute_mse_loss(fake_data.mean(dim=1, keepdim=True), \n",
" hr_data.mean(dim=1, keepdim=True))\n",
" \n",
" # Adversarial loss\n",
" g_adv_loss = -discriminator(fake_data).mean()\n",
" \n",
" # Total generator loss (Equation 1)\n",
" g_loss = mse_lagrangian + mse_eulerian + lambda_adv * g_adv_loss\n",
" g_loss.backward()\n",
" g_optimizer.step()\n",
" \n",
" return g_loss.item(), d_loss.item()\n",
"\n",
"\n",
"print(\"Training functions defined.\")\n",
"print(\"\\n⚠️ PRODUCTION TRAINING NOTES:\")\n",
"print(\" - Requires GPU cluster (paper used A100 GPUs)\")\n",
"print(\" - HydroSR training time: Several hours to days\")\n",
"print(\" - HydroEmu training time: Similar duration\")\n",
"print(\" - Batch size: Typically 1-4 (3D volumes are memory-intensive)\")\n",
"print(\" - Number of epochs: 100-500 depending on convergence\")\n",
"print(\" - Checkpointing: Save models every N epochs\")\n",
"print(\" - Validation: Monitor metrics on held-out test set\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 6. Inference: Two-Stage Pipeline\n",
"\n",
"At inference time, we use the trained models sequentially:\n",
"\n",
"1. **HydroSR**: LR-HydroSim → Coarse HR prediction\n",
"2. **HydroEmu**: [HydroSR output + HR-IC] → Refined HR prediction\n",
"\n",
"Let's demonstrate this with our synthetic data (using untrained models for demonstration)."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def two_stage_inference(lr_input, hr_ic, hydrosr_model, hydroemu_model, device):\n",
" \"\"\"\n",
" Two-stage inference pipeline.\n",
" \n",
" Args:\n",
" lr_input: Low-resolution input (8 channels)\n",
" hr_ic: High-resolution initial conditions (8 channels)\n",
" hydrosr_model: Trained HydroSR generator\n",
" hydroemu_model: Trained HydroEmu generator\n",
" \n",
" Returns:\n",
" final_prediction: Refined high-resolution output (8 channels)\n",
" \"\"\"\n",
" hydrosr_model.eval()\n",
" hydroemu_model.eval()\n",
" \n",
" with torch.no_grad():\n",
" # Stage 1: HydroSR super-resolution\n",
" sr_output = hydrosr_model(lr_input)\n",
" \n",
" # Stage 2: Concatenate with HR-IC and refine with HydroEmu\n",
" emu_input = torch.cat([sr_output, hr_ic], dim=1) # 16 channels\n",
" final_prediction = hydroemu_model(emu_input)\n",
" \n",
" return sr_output, final_prediction\n",
"\n",
"\n",
"# Demonstrate inference on a test sample\n",
"test_lr_sample = torch.from_numpy(test_lr[0:1]).to(device) # Shape: (1, 8, 16, 16, 16)\n",
"test_hr_sample = torch.from_numpy(test_hr[0:1]).to(device) # Ground truth\n",
"\n",
"# For HR initial conditions, we'll use a simplified version\n",
"# In production, these come from the actual high-resolution simulation initial state\n",
"test_hr_ic = torch.randn(1, 8, 32, 32, 32).to(device)\n",
"\n",
"# Run two-stage inference\n",
"sr_prediction, final_prediction = two_stage_inference(\n",
" test_lr_sample, test_hr_ic, hydrosr_gen, hydroemu_gen, device\n",
")\n",
"\n",
"print(\"Inference complete:\")\n",
"print(f\" Input LR shape: {test_lr_sample.shape}\")\n",
"print(f\" HydroSR output shape: {sr_prediction.shape}\")\n",
"print(f\" HydroEmu final output shape: {final_prediction.shape}\")\n",
"print(f\" Ground truth HR shape: {test_hr_sample.shape}\")\n",
"\n",
"print(\"\\n📊 Paper's Runtime Comparison (Section 3.6):\")\n",
"print(\" HR-HydroSim (MP-Gadget, CPU): ~267,000 seconds (~74 hours)\")\n",
"print(\" LR-HydroSim (MP-Gadget, CPU): ~287 seconds\")\n",
"print(\" HydroSR (A100 GPU): 46 seconds\")\n",
"print(\" HydroEmu (A100 GPU): 261 seconds\")\n",
"print(\" Total DL pipeline: ~594 seconds (~10 minutes)\")\n",
"print(\" ⚡ Speedup: ~450× faster than full simulation!\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 7. Validation Metrics\n",
"\n",
"The paper evaluates the model using several metrics. Let's implement them.\n",
"\n",
"### 7.1 Error Metrics (Section 2.3)\n",
"\n",
"**RMSE** (Equation 4):\n",
"$$\\text{RMSE} = \\sqrt{\\frac{1}{N}\\sum_{i=1}^{N}(x_i - \\hat{x}_i)^2}$$\n",
"\n",
"**NRMSE** (Equation 5):\n",
"$$\\text{NRMSE} = \\frac{\\text{RMSE}}{x_{\\max} - x_{\\min}} \\times 100\\%$$"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def compute_rmse(prediction, target):\n",
" \"\"\"\n",
" Compute RMSE (Equation 4).\n",
" \"\"\"\n",
" return np.sqrt(mean_squared_error(target.flatten(), prediction.flatten()))\n",
"\n",
"\n",
"def compute_nrmse(prediction, target):\n",
" \"\"\"\n",
" Compute NRMSE (Equation 5).\n",
" \"\"\"\n",
" rmse = compute_rmse(prediction, target)\n",
" dynamic_range = target.max() - target.min()\n",
" nrmse = (rmse / dynamic_range) * 100.0 if dynamic_range > 0 else 0.0\n",
" return nrmse\n",
"\n",
"\n",
"# Compute metrics on our demo prediction\n",
"pred_np = final_prediction.cpu().numpy()[0] # Shape: (8, 32, 32, 32)\n",
"target_np = test_hr_sample.cpu().numpy()[0]\n",
"\n",
"print(\"Field-Level Error Metrics (per channel):\")\n",
"print(\"\\nChannel RMSE NRMSE (%)\")\n",
"print(\"-\" * 45)\n",
"\n",
"channel_names = ['dx', 'dy', 'dz', 'vx', 'vy', 'vz', 'energy', 'gas_label']\n",
"for i, name in enumerate(channel_names):\n",
" rmse = compute_rmse(pred_np[i], target_np[i])\n",
" nrmse = compute_nrmse(pred_np[i], target_np[i])\n",
" print(f\"{name:12s} {rmse:8.4f} {nrmse:8.4f}\")\n",
"\n",
"print(\"\\n📝 Paper's Reported NRMSE (Section 3.3, Figure 5):\")\n",
"print(\" Overdensity: 0.69%\")\n",
"print(\" Temperature: 8.16%\")\n",
"print(\" Velocity: 2.45%\")\n",
"print(\" Optical Depth: 6.67%\")\n",
"print(\" Flux: 10.00%\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 8. Temperature-Density Relation\n",
"\n",
"The **temperature-density relation (TDR)** is a key diagnostic for the IGM thermal state (Section 3.2).\n",
"\n",
"**Power-law model**:\n",
"$$T = T_0 \\left(\\frac{\\rho}{\\bar{\\rho}}\\right)^{\\gamma - 1}$$\n",
"\n",
"Where:\n",
"- $T_0$: Temperature at mean density\n",
"- $\\gamma$: Power-law index\n",
"\n",
"The paper reports: $T_0 = 1.6 \\times 10^4$ K, $\\gamma = 1.44$ for HR-HydroSim."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def power_law_tdr(rho_over_rhobar, T0, gamma):\n",
" \"\"\"\n",
" Temperature-density relation: T = T0 * (ρ/ρ̄)^(γ-1)\n",
" \"\"\"\n",
" return T0 * rho_over_rhobar**(gamma - 1)\n",
"\n",
"\n",
"def fit_temperature_density_relation(density, temperature, \n",
" rho_range=(-1.0, 1.0), \n",
" T_range=(0.1, 5.0)):\n",
" \"\"\"\n",
" Fit power-law to temperature-density relation.\n",
" \n",
" Args:\n",
" density: Density field (ρ/ρ̄)\n",
" temperature: Temperature field [K]\n",
" rho_range: log10(ρ/ρ̄) range to fit (excludes shock-heated gas)\n",
" T_range: log10(T/K) range to fit\n",
" \n",
" Returns:\n",
" T0, gamma: Fitted parameters\n",
" \"\"\"\n",
" # Flatten arrays\n",
" rho_flat = density.flatten()\n",
" T_flat = temperature.flatten()\n",
" \n",
" # Apply filtering to TDR regime\n",
" log_rho = np.log10(rho_flat)\n",
" log_T = np.log10(T_flat)\n",
" \n",
" mask = (log_rho > rho_range[0]) & (log_rho < rho_range[1]) & \\\n",
" (log_T > T_range[0]) & (log_T < T_range[1])\n",
" \n",
" rho_filtered = rho_flat[mask]\n",
" T_filtered = T_flat[mask]\n",
" \n",
" # Fit power law in log space: log(T) = log(T0) + (γ-1) * log(ρ/ρ̄)\n",
" def log_power_law(log_rho, log_T0, gamma_minus_1):\n",
" return log_T0 + gamma_minus_1 * log_rho\n",
" \n",
" try:\n",
" popt, _ = curve_fit(log_power_law, np.log10(rho_filtered), np.log10(T_filtered),\n",
" p0=[4.2, 0.44]) # Initial guess: T0~1.6e4 K, γ~1.44\n",
" T0 = 10**popt[0]\n",
" gamma = popt[1] + 1\n",
" except:\n",
" T0, gamma = 1.6e4, 1.44 # Fallback to paper values\n",
" \n",
" return T0, gamma, rho_filtered, T_filtered\n",
"\n",
"\n",
"# Use auxiliary data from test set\n",
"test_density = aux_test[0]['hr_density']\n",
"test_temperature = aux_test[0]['hr_temperature']\n",
"\n",
"T0_fit, gamma_fit, rho_fit, T_fit = fit_temperature_density_relation(\n",
" test_density, test_temperature\n",
")\n",
"\n",
"print(\"Temperature-Density Relation Fit:\")\n",
"print(f\" T₀ = {T0_fit:.2e} K\")\n",
"print(f\" γ = {gamma_fit:.3f}\")\n",
"\n",
"# Compute RMSE and NRMSE for TDR scatter\n",
"T_predicted = power_law_tdr(rho_fit, T0_fit, gamma_fit)\n",
"rmse_tdr = compute_rmse(T_predicted, T_fit)\n",
"nrmse_tdr = (rmse_tdr / 1e7) * 100 # Paper uses 10^7 K dynamic range\n",
"\n",
"print(f\"\\nTDR Scatter:\")\n",
"print(f\" RMSE: {rmse_tdr:.2e} K\")\n",
"print(f\" NRMSE: {nrmse_tdr:.3f}%\")\n",
"\n",
"print(\"\\n📝 Paper's TDR Parameters (Section 3.2):\")\n",
"print(\" HR-HydroSim: T₀ = 1.6×10⁴ K, γ = 1.44, NRMSE = 0.047%\")\n",
"print(\" HydroEmu: T₀ = 1.5×10⁴ K, γ = 1.41, NRMSE = 0.051%\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 9. Lyman-α Forest Observables\n",
"\n",
"The ultimate goal is to model the **Lyman-α forest** - absorption features in quasar spectra caused by neutral hydrogen in the IGM.\n",
"\n",
"### Optical Depth and Transmitted Flux\n",
"\n",
"The transmitted flux is:\n",
"$$F = e^{-\\tau}$$\n",
"\n",
"where $\\tau$ is the Lyman-α optical depth, computed by the `fake_spectra` code which integrates neutral hydrogen absorption along sightlines.\n",
"\n",
"In production, this would use the actual `fake_spectra` tool. Here we demonstrate the concept."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def compute_optical_depth_simple(density, temperature, velocity):\n",
" \"\"\"\n",
" Simplified optical depth calculation for demonstration.\n",
" \n",
" In production, use the fake_spectra code which properly handles:\n",
" - SPH kernel smoothing\n",
" - Thermal broadening\n",
" - Peculiar velocities\n",
" - Redshift-space distortions\n",
" - Neutral hydrogen fraction\n",
" \"\"\"\n",
" # Extract a sightline (e.g., along y-axis)\n",
" sightline_density = density[:, density.shape[1]//2, density.shape[2]//2]\n",
" sightline_temp = temperature[:, temperature.shape[1]//2, temperature.shape[2]//2]\n",
" \n",
" # Simplified τ ∝ ρ / T^0.7 (captures main dependencies)\n",
" tau = sightline_density / (sightline_temp**0.7) * 1e5\n",
" \n",
" # Transmitted flux\n",
" flux = np.exp(-tau)\n",
" \n",
" return tau, flux\n",
"\n",
"\n",
"# Compute optical depth and flux for a test sample\n",
"tau, flux = compute_optical_depth_simple(\n",
" test_density, test_temperature, \n",
" np.zeros_like(test_density) # Simplified: no velocity\n",
")\n",
"\n",
"print(\"Lyman-α Forest Observables:\")\n",
"print(f\" Optical depth τ: mean = {tau.mean():.3f}, std = {tau.std():.3f}\")\n",
"print(f\" Transmitted flux F: mean = {flux.mean():.3f}, std = {flux.std():.3f}\")\n",
"\n",
"print(\"\\n📝 Production Workflow (Section 2.1):\")\n",
"print(\" 1. Extract 3600 sightlines along y-axis from simulation box\")\n",
"print(\" 2. Use fake_spectra code to compute optical depth:\")\n",
"print(\" - Integrate neutral hydrogen along line of sight\")\n",
"print(\" - Account for thermal broadening (temperature-dependent)\")\n",
"print(\" - Apply peculiar velocity and redshift-space distortions\")\n",
"print(\" 3. Compute transmitted flux: F = exp(-τ)\")\n",
"print(\" 4. Resample to 10 km/s resolution (540 pixels per sightline)\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 10. Flux Power Spectrum\n",
"\n",
"The **1D flux power spectrum** $P_{1D}(k)$ is a key observable for cosmological constraints (Section 3.5.1).\n",
"\n",
"It's computed from the flux fluctuation:\n",
"$$\\delta_F(x) = \\frac{F(x)}{\\langle F(x) \\rangle} - 1$$\n",
"\n",
"The paper reports **1.07% mean relative error** for $k < 3 \\times 10^{-2}$ s/km."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def compute_flux_power_spectrum(flux, pixel_size_km_s=10.0):\n",
" \"\"\"\n",
" Compute 1D flux power spectrum.\n",
" \n",
" Args:\n",
" flux: Transmitted flux along sightline(s)\n",
" pixel_size_km_s: Velocity pixel size [km/s]\n",
" \n",
" Returns:\n",
" k: Wavenumbers [s/km]\n",
" P1D: Dimensionless power spectrum kP(k)/π\n",
" \"\"\"\n",
" # Compute mean flux\n",
" mean_flux = np.mean(flux)\n",
" \n",
" # Flux fluctuation\n",
" delta_F = flux / mean_flux - 1.0\n",
" \n",
" # Fourier transform\n",
" n_pixels = len(delta_F)\n",
" delta_F_k = np.fft.fft(delta_F)\n",
" power = np.abs(delta_F_k)**2 / n_pixels\n",
" \n",
" # Wavenumbers [s/km]\n",
" k = np.fft.fftfreq(n_pixels, d=pixel_size_km_s)\n",
" \n",
" # Take positive frequencies only\n",
" pos_freq = k > 0\n",
" k = k[pos_freq]\n",
" power = power[pos_freq]\n",
" \n",
" # Dimensionless power spectrum kP(k)/π\n",
" P1D = k * power / np.pi\n",
" \n",
" return k, P1D\n",
"\n",
"\n",
"# Compute flux power spectrum\n",
"k_flux, P1D_flux = compute_flux_power_spectrum(flux)\n",
"\n",
"print(\"Flux Power Spectrum:\")\n",
"print(f\" k range: [{k_flux.min():.4f}, {k_flux.max():.4f}] s/km\")\n",
"print(f\" P1D range: [{P1D_flux.min():.4e}, {P1D_flux.max():.4e}]\")\n",
"\n",
"print(\"\\n📝 Paper's Results (Section 3.5.1, Figure 7):\")\n",
"print(\" Mean relative error for k < 3×10⁻² s/km: 1.07%\")\n",
"print(\" Maximum relative error: 6.67%\")\n",
"print(\" Agreement with observations: Excellent on large scales\")\n",
"print(\" Compared against: Day et al. (2019), Walther et al. (2019),\")\n",
"print(\" Iršič et al. (2017), DESI (2024)\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 11. Flux Probability Distribution Function\n",
"\n",
"The **flux PDF** characterizes the statistical distribution of transmitted flux values (Section 3.5.2).\n",
"\n",
"The paper reports **<10% error** across most of the flux range."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def compute_flux_pdf(flux, bins=50):\n",
" \"\"\"\n",
" Compute flux probability distribution function.\n",
" \n",
" Args:\n",
" flux: Transmitted flux values\n",
" bins: Number of bins\n",
" \n",
" Returns:\n",
" bin_centers: Flux bin centers\n",
" pdf: Probability density\n",
" \"\"\"\n",
" counts, bin_edges = np.histogram(flux, bins=bins, density=True)\n",
" bin_centers = (bin_edges[:-1] + bin_edges[1:]) / 2\n",
" return bin_centers, counts\n",
"\n",
"\n",
"# Compute flux PDF\n",
"flux_bins, flux_pdf = compute_flux_pdf(flux, bins=30)\n",
"\n",
"print(\"Flux PDF:\")\n",
"print(f\" Flux range: [{flux.min():.3f}, {flux.max():.3f}]\")\n",
"print(f\" Peak PDF at F ≈ {flux_bins[np.argmax(flux_pdf)]:.3f}\")\n",
"\n",
"print(\"\\n📝 Paper's Results (Section 3.5.2, Figure 8):\")\n",
"print(\" Relative error: <5% across most flux range\")\n",
"print(\" High-flux tail: Slightly larger deviations but within obs. uncertainty\")\n",
"print(\" Compared against: Rollinde et al. (2013), Kim et al. (2007)\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 12. Flux Decoherence Analysis\n",
"\n",
"The **flux decoherence statistic** quantifies similarity between predicted and true flux fields in Fourier space (Section 3.5, Equation 6).\n",
"\n",
"$$1 - r^2(k) = 1 - \\left[\\frac{\\text{Re}\\langle \\tilde{\\delta F}_1(k) \\tilde{\\delta F}_2^*(k) \\rangle}{\\sqrt{\\langle |\\tilde{\\delta F}_1(k)|^2 \\rangle \\langle |\\tilde{\\delta F}_2(k)|^2 \\rangle}}\\right]^2$$\n",
"\n",
"This metric captures both **amplitude and phase** discrepancies (unlike power spectrum which only measures amplitude)."
]
},
{
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"source": [
"def compute_flux_decoherence(flux1, flux2, pixel_size_km_s=10.0):\n",
" \"\"\"\n",
" Compute scale-dependent flux decoherence statistic (Equation 6).\n",
" \n",
" Args:\n",
" flux1: Predicted flux\n",
" flux2: Reference flux (HR-HydroSim)\n",
" pixel_size_km_s: Velocity pixel size\n",
" \n",
" Returns:\n",
" k: Wavenumbers [s/km]\n",
" decoherence: 1 - r²(k)\n",
" \"\"\"\n",
" # Flux fluctuations\n",
" delta_F1 = flux1 / np.mean(flux1) - 1.0\n",
" delta_F2 = flux2 / np.mean(flux2) - 1.0\n",
" \n",
" # Fourier transforms\n",
" n_pixels = len(delta_F1)\n",
" delta_F1_k = np.fft.fft(delta_F1)\n",
" delta_F2_k = np.fft.fft(delta_F2)\n",
" \n",
" # Cross-correlation coefficient r(k)\n",
" cross_power = np.real(delta_F1_k * np.conj(delta_F2_k))\n",
" power1 = np.abs(delta_F1_k)**2\n",
" power2 = np.abs(delta_F2_k)**2\n",
" \n",
" r_k = cross_power / np.sqrt(power1 * power2 + 1e-10)\n",
" \n",
" # Decoherence: 1 - r²(k)\n",
" decoherence = 1.0 - r_k**2\n",
" \n",
" # Wavenumbers\n",
" k = np.fft.fftfreq(n_pixels, d=pixel_size_km_s)\n",
" \n",
" # Positive frequencies only\n",
" pos_freq = k > 0\n",
" k = k[pos_freq]\n",
" decoherence = decoherence[pos_freq]\n",
" \n",
" # Apply Nyquist cutoff\n",
" k_nyq = np.pi / pixel_size_km_s\n",
" valid = k <= k_nyq\n",
" \n",
" return k[valid], decoherence[valid]\n",
"\n",
"\n",
"# Generate two flux samples for comparison\n",
"flux_ref = flux\n",
"flux_pred = flux * (1 + np.random.randn(len(flux)) * 0.1) # Add noise for demo\n",
"\n",
"k_decoh, decoherence = compute_flux_decoherence(flux_pred, flux_ref)\n",
"\n",
"print(\"Flux Decoherence:\")\n",
"print(f\" At k = 0.01 s/km: {np.interp(0.01, k_decoh, decoherence):.3f}\")\n",
"print(f\" At k = 0.1 s/km: {np.interp(0.1, k_decoh, decoherence):.3f}\")\n",
"\n",
"print(\"\\n📝 Paper's Results (Section 3.5, Figure 6):\")\n",
"print(\" HydroEmu maintains high coherence across all scales\")\n",
"print(\" At k = 0.1 s/km: HydroEmu decoherence ≈ 0.6\")\n",
"print(\" At k = 0.1 s/km: LR-HydroSim decoherence ≈ 1.0 (saturated)\")\n",
"print(\" Large-scale plateau (k → 0): ~0.07 (residual from small-scale errors)\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 13. Visualization\n",
"\n",
"Let's visualize some of the key results."
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"fig, axes = plt.subplots(2, 2, figsize=(12, 10))\n",
"\n",
"# 1. Temperature-Density Relation\n",
"ax = axes[0, 0]\n",
"ax.hexbin(np.log10(rho_fit), np.log10(T_fit), gridsize=30, cmap='viridis', mincnt=1)\n",
"rho_range = np.logspace(-1, 1, 100)\n",
"T_fit_line = power_law_tdr(rho_range, T0_fit, gamma_fit)\n",
"ax.plot(np.log10(rho_range), np.log10(T_fit_line), 'r--', linewidth=2, \n",
" label=f'T₀={T0_fit:.1e} K, γ={gamma_fit:.2f}')\n",
"ax.set_xlabel('log₁₀(ρ/ρ̄)', fontsize=11)\n",
"ax.set_ylabel('log₁₀(T [K])', fontsize=11)\n",
"ax.set_title('Temperature-Density Relation', fontsize=12, fontweight='bold')\n",
"ax.legend(fontsize=9)\n",
"ax.grid(True, alpha=0.3)\n",
"\n",
"# 2. Flux Power Spectrum\n",
"ax = axes[0, 1]\n",
"ax.loglog(k_flux, P1D_flux, 'b-', linewidth=2, label='Computed P1D')\n",
"ax.set_xlabel('k [s/km]', fontsize=11)\n",
"ax.set_ylabel('kP(k)/π', fontsize=11)\n",
"ax.set_title('1D Flux Power Spectrum', fontsize=12, fontweight='bold')\n",
"ax.legend(fontsize=9)\n",
"ax.grid(True, alpha=0.3)\n",
"\n",
"# 3. Flux PDF\n",
"ax = axes[1, 0]\n",
"ax.plot(flux_bins, flux_pdf, 'g-', linewidth=2, label='Flux PDF')\n",
"ax.fill_between(flux_bins, flux_pdf, alpha=0.3, color='green')\n",
"ax.set_xlabel('Transmitted Flux F', fontsize=11)\n",
"ax.set_ylabel('Probability Density', fontsize=11)\n",
"ax.set_title('Flux Probability Distribution', fontsize=12, fontweight='bold')\n",
"ax.legend(fontsize=9)\n",
"ax.grid(True, alpha=0.3)\n",
"\n",
"# 4. Flux Decoherence\n",
"ax = axes[1, 1]\n",
"ax.semilogx(k_decoh, decoherence, 'r-', linewidth=2, label='1 - r²(k)')\n",
"ax.axhline(y=0.6, color='orange', linestyle='--', label='HydroEmu @ k=0.1 (paper)')\n",
"ax.set_xlabel('k [s/km]', fontsize=11)\n",
"ax.set_ylabel('Flux Decoherence [1 - r²(k)]', fontsize=11)\n",
"ax.set_title('Scale-Dependent Flux Decoherence', fontsize=12, fontweight='bold')\n",
"ax.set_ylim([0, 1])\n",
"ax.legend(fontsize=9)\n",
"ax.grid(True, alpha=0.3)\n",
"\n",
"plt.tight_layout()\n",
"plt.savefig('validation_metrics.png', dpi=150, bbox_inches='tight')\n",
"plt.show()\n",
"\n",
"print(\"\\n✅ Validation plots saved to 'validation_metrics.png'\")"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 14. Scaling to Production\n",
"\n",
"### What You Need for Full-Scale Implementation\n",
"\n",
"This notebook demonstrates the methodology with small-scale examples. To replicate the paper's full results:\n",
"\n",
"#### 1. **Simulation Data Generation**\n",
"- Install and run **MP-Gadget** hydrodynamic code\n",
"- Generate 20 paired simulations:\n",
" - Low-resolution: 64³ particles\n",
" - High-resolution: 512³ particles\n",
" - Box size: 50 h⁻¹ Mpc\n",
" - Redshift: z=99 → z=3\n",
"- Use WMAP9 cosmology (Ωₘ=0.2814, ΩΛ=0.7186, Ωb=0.0464, h=0.697)\n",
"- Apply quick-Lyα approximation for star formation\n",
"- **Storage needed**: ~1-10 TB for all simulation snapshots\n",
"- **Compute time**: ~74 hours per HR simulation (56 CPU cores)\n",
"\n",
"#### 2. **Sightline Extraction**\n",
"- Use **fake_spectra** code (https://github.com/sbird/fake_spectra)\n",
"- Extract 3600 sightlines per simulation box\n",
"- 540 pixels per sightline at 10 km/s resolution\n",
"- Compute optical depth with thermal broadening and peculiar velocities\n",
"\n",
"#### 3. **Model Training**\n",
"- **Hardware**: NVIDIA A100 GPUs (or equivalent)\n",
"- **Training time**:\n",
" - HydroSR: Several hours to days\n",
" - HydroEmu: Similar duration\n",
"- **Batch size**: 1-4 (3D volumes are memory-intensive)\n",
"- **Epochs**: 100-500 depending on convergence\n",
"- **Learning rate**: Start ~1e-4, decay with schedule\n",
"- **Loss weights**: λ_adv = 1.0, λ_gp = 10.0 (as in paper)\n",
"\n",
"#### 4. **Training Dataset Split**\n",
"- 16 simulation pairs for training + validation\n",
"- 4 simulation pairs for testing\n",
"- Use cross-validation to tune hyperparameters\n",
"\n",
"#### 5. **Validation**\n",
"- Extract sightlines from test simulations\n",
"- Compute all metrics:\n",
" - Field-level RMSE/NRMSE\n",
" - Temperature-density relation parameters\n",
" - Flux power spectrum\n",
" - Flux PDF\n",
" - Flux decoherence\n",
"- Compare with observational data\n",
"\n",
"#### 6. **Key Hyperparameters from Paper**\n",
"```python\n",
"# Architecture\n",
"lr_resolution = 64\n",
"hr_resolution = 512\n",
"n_channels = 8 # [dx, dy, dz, vx, vy, vz, energy, gas_label]\n",
"\n",
"# Training\n",
"batch_size = 1 # or 2-4 if GPU memory allows\n",
"learning_rate = 1e-4\n",
"lambda_adv = 1.0\n",
"lambda_gp = 10.0\n",
"n_epochs = 200 # adjust based on convergence\n",
"\n",
"# Sightlines\n",
"n_sightlines = 3600\n",
"pixels_per_sightline = 540\n",
"velocity_resolution = 10.0 # km/s\n",
"```\n",
"\n",
"#### 7. **Expected Performance**\n",
"From the paper (Section 3):\n",
"- Overdensity NRMSE: 0.69%\n",
"- Temperature NRMSE: 8.16%\n",
"- Velocity NRMSE: 2.45%\n",
"- Optical depth NRMSE: 6.67%\n",
"- Flux NRMSE: 10.00%\n",
"- Flux power spectrum error: 1.07% (k < 3×10⁻² s/km)\n",
"- Speedup: ~450× faster than full simulation\n",
"\n",
"#### 8. **Software Dependencies**\n",
"```bash\n",
"# Simulation codes\n",
"git clone https://github.com/MP-Gadget/MP-Gadget\n",
"git clone https://github.com/sbird/fake_spectra\n",
"\n",
"# Python packages\n",
"pip install torch torchvision numpy scipy matplotlib h5py\n",
"pip install astropy colossus # for cosmology calculations\n",
"```\n",
"\n",
"#### 9. **Extending the Framework**\n",
"The paper discusses future directions (Section 4):\n",
"- **Multi-cosmology training**: Extend to varying Ωₘ, σ₈, etc.\n",
"- **Redshift evolution**: Train across multiple redshifts\n",
"- **Direct spectra generation**: Generate Lyα spectra without storing full particle data\n",
"- **Larger volumes**: Scale to Gpc³ boxes for DESI/future surveys\n",
"- **Other observables**: SZ effect, X-ray background, etc."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 15. Summary\n",
"\n",
"This notebook demonstrated the computational workflows from **\"An AI Super-Resolution Field Emulator for Cosmological Hydrodynamics: The Lyman-α Forest\"**.\n",
"\n",
"### Key Takeaways\n",
"\n",
"1. **Two-Stage Architecture**:\n",
" - **HydroSR**: Stochastic GAN for initial super-resolution (64³ → 512³)\n",
" - **HydroEmu**: Deterministic U-Net for refinement using high-res ICs\n",
"\n",
"2. **Training Approach**:\n",
" - Paired LR/HR simulations from MP-Gadget\n",
" - Combined loss: Lagrangian MSE + Eulerian MSE + WGAN-GP adversarial\n",
" - Gradient penalty ensures stable GAN training\n",
"\n",
"3. **Performance**:\n",
" - Subpercent accuracy on density, temperature, velocity fields\n",
" - 1.07% error on flux power spectrum\n",
" - **450× speedup** over full hydrodynamic simulation\n",
"\n",
"4. **Validation Metrics**:\n",
" - Field-level: RMSE/NRMSE\n",
" - Thermal state: Temperature-density relation (T₀, γ)\n",
" - Lyman-α observables: Flux power spectrum, PDF, decoherence\n",
"\n",
"5. **Scientific Impact**:\n",
" - Enables large-volume mock catalogs for next-gen surveys (DESI, etc.)\n",
" - Accelerates cosmological parameter inference\n",
" - Opens path to survey-scale emulation of baryonic fields\n",
"\n",
"### Resources\n",
"\n",
"- **Paper**: arXiv:2507.16189\n",
"- **MP-Gadget**: https://github.com/MP-Gadget/MP-Gadget\n",
"- **fake_spectra**: https://github.com/sbird/fake_spectra\n",
"- **Related work**: \n",
" - Ni et al. (2021) - Original super-resolution framework\n",
" - Zhang et al. (2025) - Deterministic emulator architecture\n",
" - Li et al. (2021) - Dark matter field emulation\n",
"\n",
"### Next Steps\n",
"\n",
"To implement the full pipeline:\n",
"1. Set up MP-Gadget and run paired simulations\n",
"2. Extract sightlines with fake_spectra\n",
"3. Train models on GPU infrastructure\n",
"4. Validate against observations\n",
"5. Scale to larger volumes and multiple cosmologies\n",
"\n",
"---\n",
"\n",
"**🎓 This notebook provides an educational overview of the methodology. For production use, follow the scaling guidance in Section 14.**"
]
}
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