CNN2D-GDF: Convolutional Neural Network Analysis of 2D Gaussian Density Fields

This repository contains code and methods for training convolutional neural networks (CNNs) on 2D Gaussian density field (GDF) maps to infer cosmological parameters from analytically defined power spectra. The project builds upon Villaescusa-Navarro et al. (2020) and explores the extent to which CNNs can extract information consistent with the Fisher information bound from purely Gaussian fields.

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Overview

The study focuses on the AstroNone toy dataset (as introduced in Villaescusa-Navarro et al., 2020), where 2D Gaussian density fields are generated from a simple analytical power spectrum

$$P(k) = \frac{A}{k}$$

with A as the only cosmological parameter.

The CNN model is trained to predict the value of A directly from the generated 2D Gaussian field maps. The experiment is designed to test whether deep learning can achieve the Cramér–Rao lower bound on parameter estimation.

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Data Generation and Pre-processing

• Power spectrum definition: $P(k) = A / k$
• Parameter sampling:
  $A \sim \mathcal{N}(1.0, \sigma = 0.2)$, clipped to [0.8, 1.2].
• Map generation:
  100 000 Gaussian density field maps of size $64 \times 64$ pixels were generated using the Pylians library.
• Normalization:
  • Field maps: Z-score normalization using the dataset mean and standard deviation.
  • Parameter A: Min-max scaling to [0, 1].
• Dataset split: 70 % train | 15 % validation | 15 % test.

Each map represents a realization of a Gaussian density field defined by its power spectrum amplitude A.
The goal is to recover A using supervised regression with a CNN.

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CNN Architecture

• Framework: PyTorch

• Architecture:

  • 5 convolutional layers
  • Kernel size = 4, stride = 2, padding = 1
  • Activation: LeakyReLU (α = 0.2)
  • Flatten → fully connected → single output neuron (predicting A)

• Loss function:
  Mean Squared Error (MSE)

  $$L = \frac{1}{N}\sum_{i=1}^{N}(A_\text{true} - A_\text{NN})^2$$

• Optimizer: Adam

• Regularization: Weight decay (no dropout)

• Hyperparameter optimization: Optuna

  • Algorithm: Tree-structured Parzen Estimator (TPE)
  • Parameters tuned: learning rate, weight decay, number of filters per conv layer
  • 50 trials × 200 epochs per trial

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Theoretical Validation — Fisher Matrix Formalism

To assess whether the CNN extracts the maximum possible information, its predictive uncertainty is compared with the theoretical limit derived from the Fisher information matrix.

For a single parameter A in $P(k) = A/k$:

$$F = \frac{N_{\text{modes}}}{2A^2}, \quad \sigma(A) = A \sqrt{\frac{2}{N_{\text{modes}}}}$$

Averaging over $A \in [A_{\min}, A_{\max}]$:

$$\langle \sigma(A) \rangle = \sqrt{ \frac{A_{\min}^2 + A_{\min}A_{\max} + A_{\max}^2}{1.5,N_{\text{modes}}} }$$

The CNN’s empirical prediction error on A is compared against this theoretical bound to evaluate its optimality.

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Experiments and Analysis

1. Original Gaussian Density Field Maps

• CNN trained on unfiltered 64×64 maps.
• Compared predicted errors in A with Fisher-matrix theoretical errors for $A \in [0.8, 1.2]$.
• Additional test sets of 20 000 samples each were created with fixed A = 0.82, 1.0, 1.18 to study prediction distributions.

2. Fourier-Filtered Maps (Top-Hat k Filters)

• Applied sharp cutoffs in Fourier space at:
  • $k_{\max} = 0.2,\ 0.15,\ 0.1$ ( $k_{\min} = 0$ in all cases )
• CNN retrained for each filter scale.
• Neural-network errors compared with Fisher predictions computed using Nmodes for each $k_{\max}$.

3. Gaussian-Smoothed Maps

• Applied Gaussian smoothing kernels of width 1 px and 2 px.

• CNN retrained for each case.

• Since no analytical Fisher bound exists for smoothed maps, the corresponding effective ( k_{\max} ) was inferred from the CNN’s measured error using:

  [ N_{\text{modes}} = \frac{A_{\min}^2 + A_{\min}A_{\max} + A_{\max}^2}{\sigma_A^2} ]

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Implementation Details

Component	Description
Language	Python 3
Libraries	PyTorch · NumPy · Matplotlib · Pylians · Optuna
Hardware	GPU recommended (e.g., CUDA support)
Metrics	Mean Squared Error (MSE), Predicted vs True Scatter

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Outcomes

• CNN recovers parameter A with precision close to the Fisher bound, confirming that it captures nearly all information contained in the Gaussian density fields.
• As $k_{\max}$ decreases (or smoothing increases), fewer Fourier modes → larger error in A.
• The network therefore acts as an effective information extractor consistent with theoretical expectations.

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Repository Structure

CNN2D-GDF/
├── src/
│   ├── Dataset.py ← data generation & preprocessing
│   ├── Network.py ← CNN architecture
│   ├── Training.py ← model training with Optuna tuning
│   ├── Testing.py ← model evaluation
├── results/ ← figures
├── requirements.txt
└── README.md

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References

• Villaescusa-Navarro et al. 2025
• Pylians Library
• Optuna Hyperparameter Optimization
• PyTorch Framework

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🪐 Author

Arnab Lahiry
Graduate Researcher in Astrophysics
📍 IISER Tirupati, India | CCA, Flatiron Institute, New York, USA