High-fidelity numerical simulation of partial differential equations (PDEs) remains a major computational bottleneck in many-query, real-time, and design-oriented applications. While neural operators have recently emerged as powerful surrogates by learning mappings between function spaces and approximating entire families of PDE solution operators [1–3], their standalone use often suffers from limited robustness and degraded performance under distribution shift. We propose Neural Operator Warm Starts (NOWS), a hybrid framework that integrates learned solution operators with classical iterative solvers by using neural operators to generate high-quality initial guesses for Krylov methods such as Conjugate Gradient and GMRES [4,5]. Rather than replacing established numerical methods, NOWS preserves existing discretizations and solver infrastructures, requiring no modifications to the underlying algorithms or preconditioners. The neural operator rapidly reduces the initial residual, enabling the subsequent iterative solver to converge in significantly fewer iterations while retaining its stability, accuracy, and convergence guarantees [6]. The approach is solver- and discretization-agnostic, and is compatible with finite element, finite difference, finite volume, and isogeometric analysis frameworks. Across a range of static and time-dependent PDE benchmarks, including variations in parameters, mesh resolution, and geometry, NOWS consistently reduces iteration counts by factors of two to ten and achieves end-to-end runtime reductions of up to 90\%, without loss of accuracy. These results demonstrate that neural operators can be leveraged most effectively as complementary components within rigorous numerical pipelines. NOWS provides a practical and trustworthy pathway for accelerating high-fidelity PDE simulations, bridging data-driven learning and classical computational mechanics.