Differentiable fuzzy cosmic web for field-level inference

Context. A comprehensive analysis of the cosmological large-scale structure derived from galaxy surveys involves field-level inference, which requires a forward-modelling framework that simultaneously accounts for structure formation and tracer bias. Aims. While structure formation models are well understood, the development of an effective field-level bias model remains challenging, particularly in the context of tracer perturbation theory within Bayesian reconstruction methods, which we address in this work. Methods. To bridge this gap, we developed a differentiable model that integrates augmented Lagrangian perturbation theory and non-linear, non-local, and stochastic biasing. At the core of our approach is the Hierarchical Cosmic-Web Biasing Nonlocal (HICOBIAN) model, which provides a description of a field with a positive number of tracers while incorporating a long- and short-range non-local framework via cosmic-web regions and deviations from Poissonity in the likelihood. A key insight of our model is that transitions between cosmic-web regions are inherently smooth, which we implemented using sigmoid-based gradient operations. This enables a fuzzy and, hence, differentiable hierarchical cosmic-web description, making the model well-suited for machine-learning frameworks. Results. We tested the practical implementation of this model through GPU-accelerated computations implemented in JAX , the BRIDGE code, enabling a scalable evaluation of complex biasing. Our approach accurately reproduces the primordial density field within associated error bars derived from Bayesian posterior sampling within a self-specified setting (meaning that inference is performed on data generated by the exact same forward model) as validated by two- and three-point statistics in Fourier space. Furthermore, we demonstrate that the methodology approaches the maximum encoded information consistent with Poisson noise. We also demonstrate that the bias parameters of a higher order non-local-bias model can be accurately reconstructed within the Bayesian framework for bias models with eight parameters. Conclusions. We introduce a Bayesian field-level inference algorithm that leverages the same forward-modelling framework used in galaxy, quasar, and Lyman-alpha-forest mock-catalogue generation – including non-linear, non-local and stochastic bias with redshift space distortions – providing a unified and consistent approach to the analysis of large-scale cosmic structure.