Viraj Pandya, Greg L. Bryan, T. Lucas Makinen, Austen Gabrielpillai, Christopher Carr, Drummond B. Fielding, Lars Hernquist, Matthew Ho, Kartheik Iyer, Christian Kragh Jespersen, Sophie Koudmani, Marta Laska, Pablo Lemos, Christopher C. Lovell, Lucia A. Perez, William F. Robinson, Rachel S. Somerville, Tjitske K. Starkenburg, Richard Stiskalek, Bryan Terrazas, G. Mark Voit

Semi-analytic models (SAMs) have been treating galaxy populations as dynamical systems for $\gtrsim50$ years, but their evolution equations remain poorly constrained. We introduce sapphire, a modular, automatically differentiable, GPU-accelerated SAM written from scratch in JAX. For the first time, we compute exact Jacobian matrices of our nonlinear differential equations and show that they have interpretable, non-random structures, using the Pandya et al. (2023) physical model as an initial example. Both local and global sensitivity analyses reveal that supernova energy loading is a key astrophysical parameter for galaxy evolution. We use gradient descent and Hamiltonian Monte Carlo (HMC) to perform comprehensive mock parameter recovery tests. These indicate that the z=0 stellar-to-halo-mass relation alone does not contain enough information to infer many astrophysical parameters. Using observations of star-forming galaxies from the MaNGA survey and the Behroozi et al. (2019) empirical model as one baseline, we derive multiple posteriors assuming different combinations of data, including z=0 interstellar medium gas fractions and metallicities. The inferred physical parameters suggest that galaxies self-regulate their star formation primarily through preventative rather than ejective feedback. Both Fisher and HMC forecasts demonstrate the potential of sapphire to enable precision inference for galaxy formation, but more work is needed to expand its library of models. We discuss how our unique blend of differentiability, massive GPU parallelization, numerical robustness and principled Bayesian methods sets the stage for hybrid physics-informed, data-driven discovery of galaxy formation astrophysics and cosmology. We make sapphire publicly available at https://github.com/virajpandya/sapphire.