Recent advances in machine learning and differentiable programming are changing how we think about numerical simulation in the physical sciences. In this talk, I will discuss how insights from AI have begun to fundamentally alter the design and usage of solvers for partial differential equations (PDEs), with a particular focus on differentiable simulations: numerical solvers that expose gradients with respect to inputs, parameters, and intermediate states. Differentiable PDE solvers enable the seamless integration of classical numerical methods with gradient-based learning algorithms. This capability opens up new algorithmic capabilities that go well beyond traditional “solver-as-a-black-box” usage. Gradients allow simulation components to be embedded directly into learning loops, enabling end-to-end optimization, flexible parameter estimation, and tight coupling between data-driven models and physical laws. In the context of fluid dynamics, this unlocks new opportunities for incorporating learned components (closures / solver-in-the-loop training, subgrid models, or constitutive relations) directly into Navier–Stokes solvers while maintaining physical consistency. However, full differentiability is not yet a standard feature of most mature simulation codes. I will therefore discuss practical avenues for obtaining gradients in real-world settings, and fallback strategies for non-differentiable unrolling. The talk will illustrate how these ideas translate into concrete applications. Examples include data-driven closure modeling for turbulent flows, accelerated solution strategies through learned correctors, and probabilistic generative solvers. I will emphasize how differentiable solvers do not replace classical numerical methods, but rather augment them, providing a unifying framework that bridges numerical analysis, high-performance computing, and modern machine learning. Overall, the goal of this talk is to motivate why differentiable PDE solvers represent a genuine shift in simulation paradigms, and how they are positioned to become a central building block for next-generation, hybrid physics–AI simulation frameworks.