A Review of The Impact of Deep Learning on the Analysis of Cosmological Galaxy Surveys

Detection and Classification of Astronomical Objects

A perfect task for Deep Learning!

Lanusse, Ma, Li, Collett, Li, Ravanbakhsh, Mandelbaum, Poczos (2017)

Simulated gravitational lenses used for training

Better accuracy than human visual inspection!

Metcalf, et al. (2018)

Strong lens candidates in the DESI DECam Legacy Survey

Huang et al. (2019)

Inference at the Image Level

Reiman & Göhre (2018)

AI-assisted superresolution cosmological simulations

Li, Ni, Croft, Di Matteo, Bird, Feng (2021)

x8 super-resolution

• Inputs low resolution particle displacement field, outputs samples from a distribution $p_\theta(x_{SR} | x_{LR})$

Fast, high-fidelity Lyman α forests with CNNs

Harrington, Mustafa, Dornfest, Horowitz, Lukić (2021)

are these Deep Learning models a real game-changer?

The Limitations of Black-Box Large Deep Learning Approaches

• There is a risk that a large Deep Learning model can silently fail.
  $\Rightarrow$ How can we build confidence in the output of the neural network?


• Training these models can require a very large number of simulations.
  $\Rightarrow$ Do they bring a net computational benefit?
  • In case of the super-resolution model of Li et al. (2021), only 16 full-resolution $512^3$ N-body were necessary.


• In many cases, the accuracy (not the quantity) of simulations will be the main bottleneck.
  $\Rightarrow$ What new science are these deep learning models enabling?
  • In the case of cosmological SBI, they do not help us resolve the uncertainty on the simulation model.

(Harrington et al. 2021)

Learning effective physical laws for generating cosmological hydrodynamics with Lagrangian Deep Learning

Dai & Seljak (2021)

• The Lagrangian Deep Learning approach:
  • Run an approximate Particle-Mesh DM simulation (about 10 steps)
  • Introduce a displacement $\mathbf{S}$ of particles: \begin{equation} \mathbf{S}=-\alpha\nabla \hat{O}_{G} f(\delta) \end{equation} where $\hat{O}_{g}$ is a parametric Fourier-space filter, $f$ is a parametric function of $\delta$.
    $\Rightarrow$ Respects translational and rotational symmetries.
  • Apply a non-linear function on the resulting density field $\delta^\prime$: $$F(x) = \mathrm{ReLu}(b_1 f(\delta^\prime) - b_0)$$

(Dai et al. 2018)

the forward modeling road to cosmological inference

• Instead of trying to analytically evaluate the likelihood $p(x | \theta)$, let us build a forward model of the observables.
  
  $\Longrightarrow$ The simulator becomes the physical model.


• Each component of the model is now tractable, but at the cost of a large number of latent variables.

Benefits of a forward modeling approach

• Fully exploits the information content of the data (aka "full field inference").


• Easy to incorporate systematic effects.


• Easy to combine multiple cosmological probes by joint simulations.

(Schneider et al. 2015)

...so why is this not mainstream?

The Challenge of Simulation-Based Inference

$$ p(x|\theta) = \int p(x, z | \theta) dz = \int p(x | z, \theta) p(z | \theta) dz $$ Where $z$ are stochastic latent variables of the simulator.

$\Longrightarrow$ This marginal likelihood is intractable! Hence the phrase "Likelihood-Free Inference"

Black-box Simulators Define Implicit Distributions

• A black-box simulator defines $p(x | \theta)$ as an implicit distribution, you can sample from it but you cannot evaluate it.
• Key Idea: Use a parametric distribution model $\mathbb{P}_\varphi$ to approximate the implicit distribution $\mathbb{P}$.

True $\mathbb{P}$

Samples $x_i \sim \mathbb{P}$

Model $\mathbb{P}_\varphi$

Conditional Density Estimation with Neural Networks

• I assume a forward model of the observations: \begin{equation} p( x ) = p(x | \theta) \ p(\theta) \nonumber \end{equation} All I ask is the ability to sample from the model, to obtain $\mathcal{D} = \{x_i, \theta_i \}_{i\in \mathbb{N}}$


• I am going to assume $q_\phi(\theta | x)$ a parametric conditional density


• Optimize the parameters $\phi$ of $q_{\phi}$ according to \begin{equation} \min\limits_{\phi} \sum\limits_{i} - \log q_{\phi}(\theta_i | x_i) \nonumber \end{equation} In the limit of large number of samples and sufficient flexibility \begin{equation} \boxed{q_{\phi^\ast}(\theta | x) \approx p(\theta | x)} \nonumber \end{equation}

Neural Density Estimation

Bishop (1994)

Dinh et al. 2016

• Mixture Density Networks \begin{equation} p(\theta | x) = \prod_i \pi_i(x) \ \mathcal{N}\left(\mu_i(x), \ \sigma_i(x) \right) \nonumber \end{equation}


• Conditional Normalizing Flows \begin{equation} p(\theta| x) = p_z \left( z = f^{-1}(\theta, x) \right) \left| \frac{\partial f^{-1}(\theta, x)}{\partial x} \right| \end{equation}

Example of application: Likelihood-Free parameter inference with DES SV

Jeffrey, Alsing, Lanusse (2021)

Suite of N-body + raytracing simulations: $\mathcal{D}$