Dissipative Neural Closure for Energy-Stabilized POD–Galerkin Models of Incompressible Flows

A dissipative reduced-order modeling framework is developed for the incompressible Navier–Stokes equations using a neural-network closure designed to improve the reduced dynamical robustness of POD–Galerkin systems under basis truncation. Standard POD–Galerkin reduced-order models often lose dynamical fidelity in convection-dominated flows because unresolved modal interactions alter the reduced energy-transfer balance and weaken the representation of coherent flow structures. To address this issue, a reduced-coordinate closure model is introduced and trained under a dissipativity constraint that prevents artificial energy growth in the reduced dynamics while modeling unresolved modal interactions arising from POD truncation. Under suitable reduced-system assumptions, a conditional reduced-energy estimate is derived for the closed reduced-order system. In addition, computable energy-based a posteriori estimates are developed to quantify the contributions arising from POD truncation, closure approximation, and temporal discretization. The methodology is implemented using a Taylor–Hood finite element discretization together with a semi-implicit Crank–Nicolson/Adams–Bashforth time integration scheme for the full-order model. Numerical experiments on the DFG 2D–2 benchmark (flow past a circular cylinder at Reynolds number Re = 100) indicate that standard POD–Galerkin models fail to accurately reproduce the weak oscillatory modal dynamics observed in the projected snapshot evolution despite retaining approximately 99.87% of the snapshot energy. The proposed dissipative closure suppresses transient energy amplification and produces strongly dissipative reduced dynamics under the stated reduced-system assumptions. The numerical results indicate that strict dissipativity constraints can significantly damp reduced oscillatory dynamics, revealing a fundamental tension between energetic robustness and faithful reproduction of weak coherent flow structures. This observation suggests that accurate long-time reduced-order modeling of oscillatory fluid systems may require relaxed dissipativity constraints, trajectory-aware closure strategies, or adaptive modal-transfer mechanisms. The present results should therefore be interpreted as an investigation of dissipative reduced dynamical behavior under the current numerical setting rather than a demonstration of fully developed oscillatory wake reproduction.