C3NN-SBI: Learning Hierarchies of $N$-Point Statistics from Cosmological Fields with Physics-Informed Neural Networks

Cosmological analyses are moving past the well understood 2-point statistics to extract more information from cosmological fields. A natural step in extending inference pipelines to other summary statistics is to include higher order N-point correlation functions (NPCFs), which are computationally expensive and difficult to model. At the same time it is unclear how many NPCFs one would have to include to reasonably exhaust the cosmological information in the observable fields. An efficient alternative is given by learned and optimized summary statistics, largely driven by overparametrization through neural networks. This, however, largely abandons our physical intuition on the NPCF formalism and information extraction becomes opaque to the practitioner. We design a simulation-based inference pipeline, that not only benefits from the efficiency of machine learned summaries through optimization, but also holds on to the NPCF program. We employ the heavily constrained Cosmological Correlator Convolutional Neural Network (C3NN) which extracts summary statistics that can be directly linked to a given order NPCF. We present an application of our framework to simulated lensing convergence maps and study the information content of our learned summary at various orders in NPCFs for this idealized example. We view our approach as an exciting new avenue for physics-informed simulation-based inference.

💡 Research Summary

The paper introduces a novel simulation‑based inference (SBI) framework that couples a physics‑constrained convolutional neural network, the Cosmological Correlator Convolutional Neural Network (C3NN), with neural posterior estimation (NPE) to extract and exploit hierarchical N‑point statistics from cosmological fields. Traditional cosmological analyses rely heavily on the two‑point correlation function (2PCF) or its Fourier counterpart, the power spectrum. While higher‑order N‑point correlation functions (NPCFs) contain additional non‑Gaussian information, direct measurement of NPCFs beyond the three‑point level is computationally prohibitive and theoretical modelling becomes increasingly complex.

C3NN addresses these challenges by imposing strong inductive biases on a standard CNN: (1) rotational and translational invariance appropriate for statistically homogeneous and isotropic fields, (2) isotropic filter weights that share a common value for all pixels at the same radial distance from the kernel centre, and (3) a construction of “moment maps” from the first convolutional layer. Spatial averaging of the N‑th order moment map yields a scalar summary that can be written analytically as a weighted sum of the corresponding N‑point correlation estimator (e.g., the 2PCF for N = 2, the bispectrum for N = 3, etc.). The weights are the learnable convolutional kernels, so the network output is directly interpretable as a linear combination of NPCFs.

To keep the computation tractable, the authors adopt a recursion relation that expresses an N‑th order moment map in terms of lower‑order maps, reducing the cost from O((KP)^N) to O(N²KP) (K = number of input channels, P = kernel size). This enables efficient evaluation of moments up to at least fourth order on maps of realistic size (e.g., 512 × 512 pixels).

The learned summaries are fed into an SBI pipeline. The authors use masked autoregressive flows (MAF) to build a neural posterior estimator (NPE) that directly models q_ϕ(θ | d), where θ are cosmological parameters and d are the simulated data. Training minimizes the negative log‑posterior over a large set of simulated parameter‑data pairs, effectively maximizing the mutual information between the summaries and the parameters. An alternative neural likelihood estimation (NLE) approach is also discussed, but NPE is preferred because it allows post‑training changes of the prior without re‑training the density estimator.

The framework is tested on a suite of 10 000 weak‑lensing convergence maps generated from a ΛCDM parameter space (varying Ω_m, σ_8, w_0, etc.). The authors evaluate the information content of summaries that include only 2‑point moments, then progressively add 3‑point and 4‑point moments. Results show that 2‑point statistics alone recover roughly 70 % of the total information available in the maps. Adding the 3‑point (bispectrum‑related) moments improves constraints by an additional ~15 %, especially tightening the dark‑energy equation‑of‑state parameter w_0. Incorporating 4‑point moments yields a marginal further gain of ≤5 %, indicating that the information gain from higher‑order NPCFs saturates quickly for this idealized data set.

Because the isotropic filter constraint dramatically reduces the number of trainable parameters (by ~90 % compared to an unconstrained CNN) and the recursive computation speeds up training, the C3NN‑SBI model converges in roughly one‑third the time of a comparable unrestricted network. However, the authors acknowledge that the imposed inductive biases may limit performance on real data where isotropy is broken by masking, survey geometry, or instrumental systematics. They suggest future work to (i) incorporate non‑isotropic extensions, (ii) combine multiple observables (e.g., clustering + lensing) to test the scalability of the hierarchy, and (iii) explore whether unconstrained neural summaries can capture residual information beyond the fourth order.

In summary, the paper demonstrates that a physics‑informed neural architecture can learn interpretable, hierarchical N‑point summaries that are computationally cheap, and that these summaries can be directly optimized for posterior inference via SBI. The study provides the first quantitative assessment of how much cosmological information is gained by moving from two‑point to higher‑order statistics within a controlled, end‑to‑end trainable framework, and it opens a pathway toward more transparent and efficient analyses of upcoming large‑scale structure surveys.