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.