A Differentiable Programming Framework for Accurate and Stable Reduced-Order Modeling of Chaotic Flows
Classical Proper Orthogonal Decomposition (POD)-based Galerkin projection models of chaotic flows typically require a large number of modes as well as stabilization or closure terms to achieve adequate accuracy and long-term stability. We present a novel differentiable programming framework that stabilizes low-rank POD-Galerkin models without increasing the number of modes or introducing additional closure terms, thereby delivering both high efficiency and high accuracy. Model stabilization is achieved by tuning the linear and quadratic tensors in the POD-Galerkin using differentiable programming, trained on short-term trajectory data. A key finding of this study is that a purely point-wise trajectory-based loss function yields poor long-term accuracy for chaotic systems. In contrast, a hybrid loss function that combines trajectory error with a physics-based conservation-of-energy term provides superior long-term performance. We demonstrate the approach on a chaotic lid-driven cavity flow at Re = 30,000. The stabilized ROM achieves an order-of-magnitude reduction in computational cost compared with the classical POD-Galerkin method: it remains accurate and stable with only 20 modes, whereas the classical ROM requires 80 POD modes.