On closures for reduced order models – A spectrum of first-principle to machine-learned avenues

The paper explicitly defines the closed ROM with a closure term da/dt = B + L a + a^T N a + C (its Eq. 54) and then formalizes two least-squares formulations: trajectory regression (constrained fit of aROM to aFOM by solving the closed ROM; its Eq. 57) and model regression (unconstrained fit of a postulated closure CROM to filtered labels CFOM; its Eq. 58). The candidate solution restates these two formulations precisely in discrete time with an explicit IVP constraint for trajectory regression and a direct least-squares fit for model regression, matching the paper’s content; minor differences are present (e.g., allowing a purely time-indexed CROM), but do not contradict the paper’s definitions .