Neural Implicit Flow: A mesh-agnostic representation paradigm for spatio-temporal fields

Fluid dynamics exhibits complex, multi-scale spatial structure, chaotic dynamics in time, and bifurcation in the relevant parameters. Among these challenges, spatial complexity is the major barrier for modeling and control of fluid dynamics, which motivates the need for dimensionality reduction. Existing paradigms, such as proper orthogonal decomposition or convolutional autoencoders, both struggle to accurately and efficiently represent flow structures for problems requiring variable geometry, non-uniform grid resolution (e.g., wall-bounded flows, flow phenomenon induced by small geometry features), adaptive mesh refinement, or parameter-dependent meshes. To resolve these difficulties, we propose Neural Implicit Flow (NIF) as a general framework that enables a compact and flexible dimension reduction of large-scale, parametric, spatial-temporal data into mesh-agnostic fixed-length representations. This work complements existing meshless methods, e.g., physics-informed neural networks, and we focus specifically on obtaining reduced coordinates where modeling and control tasks may be performed more efficiently. We apply our mesh-agnostic approach to several fluid flows, including flow past a cylinder, sea surface temperature data, and 3D homogeneous isotropic turbulence. In these examples, we demonstrate the utility of NIF for parametric surrogate modeling, efficient differential query in space, learning non-linear manifolds, and the interpretable low-rank decomposition of fluid flow data.