\begin{algorithm}[htb]
\caption{Forward propagation for each convolutional 3D layer. $M_w$, $\Sigma_w$ are the means and variances of each layer, $H$ is the input layer, and NF(·) is the masked
RealNVP normalising flow applied over samples initially drawn from a Gaussian distribution $q$. $D_f$ is the number of filters for each kernel. $\odot$ corresponds to element-wise multiplication.} 
\label{alg:conv_bnn3d}
\begin{algorithmic}
\State $\text{Input: feature vector of the previous layer (minibatch)}$
\State $H \gets \text{Input conv3D-layer (minibatch)}$
    \State $\*z_0 \sim q(\*z_0)$
    \State $\*z_{T_f} = \text{NF}(\*z_0)$
    \State $\*M_h = \*H * (\*M_w \odot \text{reshape}(\*z_{T_f}, [1,1, 1, D_f]))$
    \State $\*V_h = \*H^2 * \!\Sigma_w$
    \State $\*E \sim \mathcal{N}(0, 1)$
    \State return $\*M_h + \sqrt{\*V_h} \odot \*E$
    \State$\text{Output: sample of feature vector according to Eq.~\ref{mnfpos3d}}$
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
[htb]
\caption{Forward propagation for each convolutional 3D layer. $M_w$, $\Sigma_w$ are the means and variances of each layer, $H$ is the input layer, and NF(·) is the masked
RealNVP normalising flow applied over samples initially drawn from a Gaussian distribution $q$. $D_f$ is the number of filters for each kernel. $\odot$ corresponds to element-wise multiplication.}
\begin{algorithmic}
\State $\text{Input: feature vector of the previous layer (minibatch)}$
\State $H \gets \text{Input conv3D-layer (minibatch)}$
\State $\*z_0 \sim q(\*z_0)$
\State $\*z_{T_f} = \text{NF}(\*z_0)$
\State $\*M_h = \*H * (\*M_w \odot \text{reshape}(\*z_{T_f}, [1,1, 1, D_f]))$
\State $\*V_h = \*H^2 * \!\Sigma_w$
\State $\*E \sim \mathcal{N}(0, 1)$
\State return $\*M_h + \sqrt{\*V_h} \odot \*E$
\State$\text{Output: sample of feature vector according to Eq.~\ref{mnfpos3d}}$
\end{algorithmic}
\end{algorithm}
{
    "title": "bayesian deep learning for cosmic volumes with modified gravity",
    "id": "2309.00612",
    "abstract": "the new generation of galaxy surveys will provide unprecedented data allowing us to test gravity at cosmological scales. a robust cosmological analysis of the large-scale structure demands exploiting the nonlinear information encoded in the cosmic web. machine learning techniques provide such tools, however, do not provide a priori assessment of uncertainties. this study aims at extracting cosmological parameters from modified gravity (mg) simulations through deep neural networks endowed with uncertainty estimations. we implement bayesian neural networks (bnns) with an enriched approximate posterior distribution considering two cases: one with a single bayesian last layer (bll), and another one with bayesian layers at all levels (fullb). we train both bnns with real-space density fields and power-spectra from a suite of 2000 dark matter only particle mesh $n$-body simulations including modified gravity models relying on mg-picola covering 256 $h^{-1}$ mpc side cubical volumes with 128$^3$ particles. bnns excel in accurately predicting parameters for $\\omega_m$ and $\\sigma_8$ and their respective correlation with the mg parameter. we find out that bnns yield well-calibrated uncertainty estimates overcoming the over- and under-estimation issues in traditional neural networks. we observe that the presence of mg parameter leads to a significant degeneracy with $\\sigma_8$ being one of the possible explanations of the poor mg predictions. ignoring mg, we obtain a deviation of the relative errors in $\\omega_m$ and $\\sigma_8$ by at least $30\\%$. moreover, we report consistent results from the density field and power spectra analysis, and comparable results between bll and fullb experiments which permits us to save computing time by a factor of two. this work contributes in setting the path to extract cosmological parameters from complete small cosmic volumes towards the highly nonlinear regime.",
    "categories": "astro-ph.co astro-ph.im cs.lg stat.ml",
    "doi": "10.1051/0004-6361/202347929",
    "created": "2023-09-01",
    "updated": "2024-02-12",
    "authors": [
        "jorge enrique garc\u00eda-farieta",
        "h\u00e9ctor j hort\u00faa",
        "francisco-shu kitaura"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2309.00612"
}
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