Differentiable Fuzzy Cosmic-Web for Field Level Inference
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. While structure formation models are well-understood, the development of an effective field-level bias model remains challenging within Bayesian reconstruction methods, which we address in this work. To bridge this gap, we have developed a differentiable model that integrates augmented Lagrangian perturbation theory, nonlinear, nonlocal, and stochastic biasing. At the core of our approach is the HICOBIAN model, which provides a description of a field with a positive number of tracers while incorporating a long- and short-range nonlocal framework 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 implement 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. We test the practical implementation of this model through GPU-accelerated computations implemented in JaX, the BRIDGE code, enabling 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 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 nonlocal bias model can be accurately reconstructed within the Bayesian framework for bias models with eight parameters.