Analytical Surrogates for Incompressible Flow Dynamics via Physics-Based Optimization

We present a parametric analytical framework for approximating two-dimensional incompressible Navier–Stokes solutions using structured, non-orthogonal functional analysis. The velocity and pressure fields are represented by explicit analytical expressions combining temporal exponential decay, trigonometric components, and low-degree polynomial terms in space, resulting in a finite-dimensional model with a moderate number of free parameters. Unlike classical spectral or Galerkin methods, the proposed approach does not rely on orthogonality, completeness, or projection onto predefined bases.

The unknown coefficients are determined by minimizing a global residual-based functional composed of the squared Navier–Stokes equations evaluated throughout the domain, together with penalized boundary constraints. This formulation enforces physical consistency in the interior while using boundary information as the only prescribed data. The residual minimization acts as an implicit regularization mechanism, despite the absence of a weak formulation or mesh-based approximation of the solution fields.

Numerical integration of the residual functional is performed on an unstructured finite element mesh generated in FreeFEM, which is employed solely as a quadrature tool. The analytical representation remains global and continuous over the domain. A genetic algorithm is used to solve the resulting non-convex optimization problem, serving as a robust global optimizer rather than an intrinsic component of the methodology.

The framework is assessed using two complementary test cases: a controlled verification problem admitting an exact analytical solution, and the classical Taylor–Green vortex benchmark, widely used for validation of Navier–Stokes solvers. Results demonstrate that physically consistent approximations can be reconstructed using boundary information alone, even for nontrivial vortical dynamics.

The proposed approach is positioned as a complementary tool bridging analytical modeling, numerical fluid dynamics, and optimization-based methods, and is particularly suited as a low-cost analytical surrogate or warm-start strategy for data-driven and physics-informed solvers in fluid dynamics applications.