A data free neural operator enabling fast inference of 2D and 3D Navier Stokes equations

Ensemble simulations of high-dimensional flow models (e.g., Navier-Stokes-type PDEs) are computationally prohibitive for real-time appli cations. Neural operators enable fast inference but are limited by costly data req uirements and poor generalization to 3D flows. We present a data-free operator n etwork for the Navier-Stokes equations that eliminates the need for paire d solution data and enables robust, real-time inference for large ensemble for ecasting. The physics-grounded architecture takes initial and boundary conditio ns as well as forcing functions, yielding solutions robust to high variability a nd perturbations. Across 2D benchmarks and 3D test cases, the method surpasses prior n eural operators in accuracy and, for ensembles, achieves greater efficie ncy than conventional numerical solvers. Notably, it delivers accurate solutions of the three-dimensional Navier-Stokes equations--a regime not previously demonstr ated for data-free neural operators. By uniting a numerically grounded archit ecture with the scalability of machine learning, this approach establishes a pra ctical pathway toward data-free, high-fidelity PDE surrogates for end-to-end sci entific simulation and prediction. Solving PDEs efficiently and accurately is one of the central interests for scienc e and engineering. In addition, when dealing with various boundary conditions, initial con ditions, or external forcing terms of PDEs in fields such as fluid mechanics [1-3], materials science [4, 5], weather forecasting [6, 7], and design optimization [8, 9], P DEs are often required to be solved repeatedly. However, conventional numeric al solvers become prohibitively expensive in such settings, particularly for three-dimensional incompressible Navier-Stokes equations (NSEs) [10, 11]. This is because these s olvers rely on spatial-temporal discretization and iterative treatment of nonline ar terms, while performing time marching that demands substantial memory and computation. Moreover, they are not well suited for solving large ensembles of scenarios simu ltaneously, such as those required for uncertainty quantification or design explora tion. The resulting computational time, coupled with the need for extensive sampling in e nsemble or probabilistic simulations, constitutes a critical bottleneck [7, 12].