• Deep Belief Network
  (Hinton et al. 2006)


• Variational AutoEncoder
  (Kingma & Welling 2014)


• Generative Adversarial Network
  (Goodfellow et al. 2014)


• Wasserstein GAN
  (Arjovsky et al. 2017)

• Training set: GalSim COSMOS HST/ACS postage stamps
  
  • 80,000 deblended galaxies from I < 25.2 sample
  • Drawn on 128x128 stamps at 0.03 arcsec resolution
  • Each stamp comes with:
    • PSF
    • Noise power spectrum
    • Bulge+Disk parametric fit



• Auto-Encoder model:
  • Deep residual autoencoder:
    7 stages of 2 resnet blocs each
  • Dense bottleneck of size 32.
  • Outputs positive, noiseless, deconvolved, galaxy surface brightness.

• $p_\theta(x)$ is explicit! We get a number out.


• We can train the model to learn a distribution of isolated galaxy images.


• We can then evaluate its gradient $\frac{\color{orange} d \color{orange}\log \color{orange}p\color{orange}(\color{orange}x\color{orange})}{\color{orange} d \color{orange}x}$.

• $P$ is the convolution with the instrumental response


• $A_i$ are channel-wise galaxy SEDs, $S_i$ are the morphology models


• $\mathbf{\Sigma}$ is the noise covariance


• $\log p_\theta$ is a PixelCNN prior


• $f_i$ and $g_i$ are arbitrary additional non-smooth consraints, e.g. positivity, monotonicity...

• Measure the ellipticity $\epsilon = \epsilon_i + \gamma$ of all galaxies
  $\Longrightarrow$ Noisy tracer of the weak lensing shear $\gamma$


• Compute summary statistics based on 2pt functions,
  e.g. the power spectrum


• Run an MCMC to recover a posterior on model parameters, using an analytic likelihood $$ p(\theta | x ) \propto \underbrace{p(x | \theta)}_{\mathrm{likelihood}} \ \underbrace{p(\theta)}_{\mathrm{prior}}$$

• Instead of trying to analytically evaluate the likelihood, let us build a forward model of the observables.


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

A non-exhaustive list of methods:
• Hamiltonian Monte-Carlo
• Variational Inference
• MAP+Laplace
• Gold Mining
• Dimensionality reduction by Fisher-Information Maximization

• The numerical scheme:
  
  
  • Estimate the density of particles on a mesh
    => compute gravitational forces by FFT
  
  
  • Interpolate forces at particle positions
  
  
  • Update particle velocity and positions, and iterate


• Fast and simple, at the cost of approximating short range interactions.

											                  import tensorflow as tf
											                  import flowpm
											                  # Defines integration steps
											                  stages = np.linspace(0.1, 1.0, 10, endpoint=True)

											                  initial_conds = flowpm.linear_field(32,        # size of the cube
											                                                      100,       # Physical size
											                                                      ipklin,    # Initial powerspectrum
											                                                      batch_size=16)

											                  # Sample particles and displace them by LPT
											                  state = flowpm.lpt_init(initial_conds, a0=0.1)

											                  # Evolve particles down to z=0
											                  final_state = flowpm.nbody(state, stages, 32)

											                  # Retrieve final density field
											                  final_field = flowpm.cic_paint(tf.zeros_like(initial_conditions),
											                                                 final_state[0])

											                  with tf.Session() as sess:
											                      sim = sess.run(final_field)
											              	

• Seamless interfacing with deep learning components
• Mesh TensorFlow implementation for distribution on supercomputers