Modular operator superposition (MOS): A physics-guided machine learning framework for addressing the curse of dimensionality and multiscale challenges in computational fluid dynamics

We introduce Modular Operator Superposition (MOS), a physics-guided and AI-augmented framework for efficient, scalable, and generalizable flow field modeling in high-dimensional and multiscale fluid systems. Rather than globally resolving flow fields via mesh-based discretization, MOS decomposes the system into physically meaningful flow primitives, each represented by a reusable modular operator. These operators are trained offline using a parameterized physics-informed neural network (P-PINN) in a single pre-processing step, and later composed through a physics-guided superposition strategy to approximate the full system-level mapping. The core advantage of MOS lies in its modularization strategy. By learning only small-scale flow primitives offline, MOS reduces the training cost to a fixed, minimal investment independent of system-level complexity. In the online stage, MOS dynamically solves for primitive-level interactions for any specific configuration of a system, then reconstructs the global flow field through the superposition of modular outputs. This two-stage online process, comprising both solving and inference, enables scalable and generalizable predictions. As a result, MOS addresses the curse of dimensionality by reducing high-dimensional systems to tractable compositions of modular operators and overcomes multiscale challenges through a scale-adaptive operator assembly that flexibly resolves flow features with minimal overhead. We demonstrate MOS for static and dynamic arrays of up to $15,000$ cylinders in a channel cross-flow (corresponding to roughly ${10}^5$ input parameters). All of these configurations are solved using a shared single-cylinder cross-flow modular operator, trained offline in 30 hours using a data-free, physics-informed machine learning strategy. In the online stage, MOS achieves end-to-end flow field prediction at 3 to 5 orders of magnitude speedup over conventional numerical solvers, while maintaining high fidelity ($R^2>0.85$ for all cases). Moreover, the MOS solution format requires 3 to 5 orders of magnitude lower memory usage than conventional numerical outputs. Once solved, the solution can be queried in real-time to infer flow variables at arbitrary spatial resolutions or scattered points, enabling flexible and efficient visualization across scales. Additional tests indicate that MOS remains robust to polydispersity and translational/rotational motion of the cylinders.