The small-scale physics of inflation can leave unique observational signatures in the gravitational wave background and may also generate primordial black holes as a dark matter candidate. These phenomena often involve a significant enhancement of inflationary fluctuations, potentially leading to the breakdown of standard perturbation theory. In this talk, I will discuss how lattice simulations provide a crucial tool for addressing these challenges. Focusing on single-field inflationary models with deviations from slow-roll, I will demonstrate how nonlinear physics can significantly alter the inflationary dynamics and their predictions, even in regimes where perturbation theory is typically assumed valid. This highlights the indispensable role of lattice simulations in probing inflation with current and future gravitational wave experiments.