Having an unstable computational problem, e.g., transient Navier-Stokes equations with variable Reynolds number, we employ Petrov-Galerkin formulation with the optimal test functions to stabilize the computational problem in every time step. The optimal test functions can be represented by the matrix of coefficients. Namely, we replace Bx=f with BTWx=f in every time step. Solving the modified system with the Petrov-Galerkin method results in the stabilization of the transient simulation. However, this method has two drawbacks. The first one is that the matrix of coefficients W of the optimal test functions changes with the Reynolds number, and the cost of computing matrix W is large. The second drawback is that the matrix of coefficients of the optimal test functions can be dense, and the cost of solving the modified system can be larger than the cost of solving the original system. To overcome them, we propose two solutions. First, we train the Deep Neural Network the matrix of coefficients of the optimal test functions as a function of the Reynolds number. In this way, we can obtain the stabilized system fast just by providing the Reynolds number to the neural network. Second, we store the matrix of coefficients of the optimal test functions in a hierarchical matrix manner. Due to efficient compression, we can solve the system of equations fast, including the automatic stabilization with the Petrov-Galerkin method and optimal test functions. This work is an extension of the stationary case employed for the advection-dominated diffusion equations [1] using an operator splitting solver for Navier-Stokes [2]. This Project has received funding from the European Union’s Horizon Europe research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101119556. [1] T. Służalec, M. Dobija, A. Paszyńska, I. Muga, M. Łoś, M. Paszyński, Automatic stabilization of finite-element simulations using neural networks and hierarchical matrices, Computer Methods in Applied Mechanics and Engineering, 411: 116073, 2023. https://doi.org/10.1016/j.cma.2023.116073. [2] M. Łoś, I. Muga, J. Muñoz-Matute, M. Paszyński, Isogeometric residual minimization (iGRM) for non-stationary Stokes and Navier–Stokes problems, Computers & Mathematics with Applications, 95: 200-214, 2021, https://doi.org/10.1016/j.camwa.2020.11.013.