In this talk, I will present recent results from a new code for simulating multifield inflationary models on a lattice. Focusing on axion-U(1) inflation, we accurately characterize the rich phenomenological features predicted by this model, including a modified small-scale power spectrum, a blue-tilted, equilateral-peaked bispectrum with a full shape that has a nontrivial, non-separable form, and higher-order correlation functions that violate parity. The output of our simulations consists of realizations of the primordial curvature field imbued with this nontrivial hierarchy of N-point statistics. Crucially, such realizations cannot be generated with standard techniques for primordial non-Gaussian initial conditions, making our approach essential for comprehensive analysis using field-level and simulation-based inference. Using the output of our inflation simulations as initial conditions for N-body simulations, we effectively simulate the entire history of the Universe from inflation to late-time large-scale structure, characterizing the observable predictions of axion-U(1) inflation for galaxy surveys. These results establish simulating inflation as an indispensable tool for broadening the space of testable, early-universe theories.