OCCAMS: A Multi-Task Learning-Based Investigation of the CMB Power Spectra

Ankit Mandal; Rahul Shah; Md. Wali Hossain; Somasri Sen; Anjan A Sen; Supratik Pal

astro-ph.COOct 29, 2026

OCCAMS: A Multi-Task Learning-Based Investigation of the CMB Power Spectra

Authors: Ankit Mandal, Rahul Shah, Md. Wali Hossain, Somasri Sen, Anjan A Sen, Supratik Pal

Organizations: Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700 108, India · Department of Physics, Jamia Millia Islamia, New Delhi 110025, India · Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India

Abstract

How many latent variables are needed to fit the cosmic microwave background (CMB) temperature and polarisation power spectra when a neural network is given no cosmological model, and to what extent do they map onto the standard cosmological parameters? We use multi-task learning with variational autoencoders trained on CLASS-generated spectra (TT, TE, EE) over $2\le\ell\le2500$ . No cosmological model is built into the network, and the number of latent variables, $L$ , is varied over $L=5$ , $6$ and $7$ . Their fits are compared to the Planck 2018 data using $\hatχ^2$ , AIC, BIC and Bayesian evidence. Their correspondence with cosmological parameters is measured with normalised mutual information post-training. The loss weights are set within a single training run by uncertainty weighting, with the weight of the disentanglement term either fixed (semi-automated, SA) or learned (completely automated, CA). The reconstruction loss is split over multipole ranges in three configurations that separate the roles of temperature and polarisation and of the low- and high- $\ell$ ranges. In four of the six combinations of configuration and strategy, the fit improves from $L=5$ to $L=6$ and not beyond, and $L=6$ is the best, or close to the best, in five of the six cases. So, six latent variables are needed, as many as the free parameters of the standard, model-dependent approach. SA gives clearer disentanglement, while CA needs no a priori choice of the disentanglement weight and has the higher evidence at $L=5$ and $6$ : the two strategies are points on a trade-off between interpretability and fit to the data. Notably, the latent variables for $τ$ and $A_s$ share almost no information in any model, consistent with large-scale polarisation separating the two, as in standard analyses. A deep-learning approach can thus give a compact, data-driven parametrisation of the CMB spectra.

Figures & tables

Panel of Figure 1 — Figure 1: The VAE network architecture. The input spectra are compressed via successive 1D convolutions into a latent distribution defined by $\mu$ and $\log\sigma^{2}$ . A stochastic sample $z$ is drawn and subsequently upsampled via transposed convolutions to reconstruct the original spectra.

Parameter	Sampling Range
$\omega_{\mathrm{b}}$	$[0.020,0.024]$
$\omega_{\mathrm{cdm}}$	$[0.10,0.13]$
$h$	$[0.62,0.80]$
$\tau$	$[0.01,0.13]$
$n_{\mathrm{s}}$	$[0.92,1.01]$
$\ln(10^{10}A_{\mathrm{s}})$	$[2.90,3.18]$

Table 1: Uniform sampling ranges used to generate the training data set via a Latin hypercube. Following Piras et al. (2025) , the ranges span approximately

10\sigma

around the Planck 2018 baseline best-fit values. There are two exceptions to these bounds: the lower limit for the optical depth (

\tau

) is restricted to

0.01

to prevent non-physical negative values, and the upper limit for the reduced Hubble constant (

h

) is broadened to encompass the local distance-ladder measurement from the SH0ES collaboration ( Riess et al., 2022 ) .

Configuration Name	Reconstruction Terms $\mathcal{T}$
lowTTTEEE+full	$\mathcal{T}=\{\mathcal{L}_{\text{low}\,\ell},\ \mathcal{L}_{\text{full}\,\ell}\}$
lowTTTEEE+high	$\mathcal{T}=\{\mathcal{L}_{\text{low}\,\ell},\ \mathcal{L}_{\text{high}\,\ell}\}$
lowTT-TEEE+high	$\mathcal{T}=\{\mathcal{L}_{\text{lowTT}\,\ell},\ \mathcal{L}_{\text{lowTEEE}\,\ell},\ \mathcal{L}_{\text{high}\,\ell}\}$

Table 2: Summary of the three loss function configurations considered here.

Panel of Figure 2 — Figure 2: The end-to-end Multi-Task Learning VAE pipeline for CMB parametrisation, shown for the CA strategy. Top: The theoretical $\Lambda$ CDM model is simulated via CLASS to generate target temperature and polarisation spectra. Middle (violet dashed box): These spectra are compressed by the encoder into an $L$ -dimensional probabilistic latent space, before being reconstructed by the decoder. Bottom (red dashed box): The extraction of data to compute the individual loss terms during training. A trainable parameter $\sigma_{i}$ is attached to each term. Backpropagation (solid red curves) updates the network parameters ( $\phi,\theta$ ) and the $\sigma_{i}$ together, so that the loss weights are set in a single training run. In SA strategy, $w_{d}$ is fixed and the disentanglement loss term is not assigned any $\sigma_{i}$ .

Figure 3: Absolute reconstruction residuals $|\Delta D_{\ell}|=|D_{\ell}^{\rm VAE}-D_{\ell}^{\rm true}|$ of the Planck 2018 best-fit $\Lambda$ CDM TT (top), TE (middle) and EE (bottom) spectra. Each pair of columns shows one configuration, with SA (left) and CA (right). The latent dimensions are $L=5$ (blue), $L=6$ (orange) and $L=7$ (green).

Figure 4: Normalised MI between each $\Lambda$ CDM parameter (rows) and each latent variable (columns) for all 18 trained models. The three blocks are the configurations, each shown for the SA and CA strategies, and the columns of panels correspond to $L=5,6,7$ . The number above each panel is the disentanglement score $S$ . Solid outline: parameter-latent pair assigned by the matching, with normalised MI of at least 0.10; dashed outline: assigned pair below the threshold of 0.10, counted as not captured. All values are on a common colour scale that follows a power law with exponent 0.6 so that small values can be told apart.

Configuration	Strategy	$\mathbf{L}$	$\mathbf{\hat{\chi}^{2}}$	$\mathbf{\Delta}$ AIC	$\mathbf{\Delta}$ BIC	$\mathbf{\ln B}$	Score
lowTTTEEE+full	SA	5	0.9630	1.1	0.0	0.3	0.619
		6	0.9607	0.7	4.0	0.0	0.737
		7	0.9578	0.0	7.8	0.4	0.781
	CA	5	0.9575	0.0	0.0	0.0	0.388
		6	0.9603	2.8	7.2	3.5	0.144
		7	0.9621	4.9	13.7	5.5	0.399

Table 3: Goodness of fit and model comparison for all 18 models, using the MCMC chains and the Plik_lite likelihood.

\Delta

AIC,

\Delta

BIC and

\ln B

are measured from the best

L

within each configuration and strategy, and the lowest values are highlighted in bold. Score is the disentanglement score

S_{\rm total}

of equation ( 7 ), computed from the normalised MI matrices of Fig. 4 .

Configuration	$L=5$	$L=6$	$L=7$
lowTTTEEE+full	$+4.3$	$+0.6$	$-1.0$
lowTTTEEE+high	$+5.6$	$+2.5$	$-0.5$
lowTT-TEEE+high	$+10.2$	$+3.4$	$-0.8$

Table 4: Evidence comparison of the two disentanglement strategies at fixed latent dimension:

\ln B_{\rm CA,SA}=\ln Z_{\rm CA}-\ln Z_{\rm SA}

for the same configuration. Positive values favour CA and negative values favour SA.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 5 — Figure 5: Visualisation of the disentanglement penalty calculation during a training epoch. The network multiplies the transposed standardised parameter batch ( $\tilde{\Theta}^{T}$ ) by the standardised latent batch ( $\tilde{Z}$ ). Each element $R_{i,j}$ is the squared correlation between parameter $i$ (e.g. $\tau$ ) and latent $j$ . Normalising each column of $R$ to sum to one gives $\mathbf{M}$ , from which the three terms of $\mathcal{L}_{\text{dis}}$ are computed: purity and strict from the columns of $\mathbf{M}$ , and collision from the off-diagonal elements of $\mathbf{M}^{T}\mathbf{M}$ . Minimising these terms makes each latent follow only a few parameters, and different latents follow different ones.

Configuration	Strategy	$L$	$\mathbf{w_{l}\,(10^{4})}$	$\mathbf{w_{h}\,(10^{2})}$	$\mathbf{\beta}$	$\mathbf{w_{d}}$
lowTTTEEE+full	SA	5	0.168	0.283	0.253	5
		6	0.994	1.16	0.209	5
		7	0.973	1.21	0.238	5
	CA	5	0.346	0.225	0.238	0.810
		6	1.28	1.21	0.191	0.587
		7	1.69	1.27	0.192	0.446

Appendix

Table 5: Loss weights at the end of training for all 18 models.

w_{l}

and

w_{h}

are the weights of the low-

\ell

and high-

\ell

reconstruction terms,

\beta

is that of the KL term, and

w_{d}

is that of the disentanglement term. For lowTT-TEEE+high the

w_{l}

column lists

(w_{\rm lowTT},w_{\rm lowTEEE})

, and for lowTTTEEE+full the

w_{h}

column holds

w_{\rm full}

Figure 6: Sensitivity of the disentanglement score to the MI threshold for the three configurations (solid: SA, dashed: CA; colours:

L

). The dotted line marks the nominal threshold of 0.10 used in the main analysis.

Figure 7: Posterior distributions of the latent variables for the lowTTTEEE+high models with

L=5

: SA (top) and CA (bottom). Contours enclose 68 and 95 per cent of the posterior.

Figure 8: Posterior distributions of the latent variables for the lowTTTEEE+high models with

L=6

: SA (top) and CA (bottom). Contours enclose 68 and 95 per cent of the posterior.

Figure 9: Posterior distributions of the latent variables for the lowTTTEEE+high models with $L=7$ : SA (top) and CA (bottom). Contours enclose 68 and 95 per cent of the posterior.

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Emulating Cosmic Structure Formation with a Lagrangian Neural Cellular Automaton

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Department of Physics, Stanford University, Stanford, CA 94305, USA · Kavli Institute for Particle Astrophysics and Cosmology, Stanford · Leinweber Institute for Theoretical Physics, Stanford

Neural Surrogate Modeling Neural Cellular Automata

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Field-level inference of cosmological initial conditions from galaxy surveys requires a forward model that is simultaneously accurate in the non-linear regime, computationally efficient, and fully differentiable. Traditional N-body simulations are accurate but computationally prohibitive for iterative inference, while approximate solvers like Lagrangian Perturbation Theory (LPT) fail to capture the knotty halo-forming dynamics of the cosmic web at late times. We introduce the Lagrangian Neural Cellular Automaton (LNCA), a hybrid deep learning framework that can be applied to emulate structure formation as a local, iterative dynamical process on a comoving lattice. Unlike standard Eulerian Convolutional Neural Networks (CNNs) which map fixed density fields, the LNCA operates in the Lagrangian frame, advecting the computational graph itself to follow the flow of mass. By training the network to learn only the residual displacement corrections to the Zeldovich approximation, we achieve high-fidelity emulation of the non-linear physics while guaranteeing accuracy at large scales. We further constrain our model to produce complete trajectories, not just final states, by adopting an equivariant cellular automaton architecture, which recurrently iterates on its internal states to yield a dynamic history. The resulting model is strictly local, translationally and rotationally equivariant, and naturally supports continuous time integration, making it a reliable differentiable forward model for reconstructing the initial conditions of the universe from lightcone data. Our trained model supports percent-level precision in the power and cross spectra well into the non-linear regime (

k \lesssim 0.5 \, h \text{Mpc}^{-1}

), while requiring

\sim10^4

times fewer learned parameters than comparable models which take the form of an interpretable internal dynamic rule set.

Abstract

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