Construction of stable difference schemes using a generative model

High-fidelity turbulent flow simulations require long-time calculations with non-dissipative schemes to accurately determine high-order flow statistics. Stable boundary treatments are key to these calculations over non-trivial domains. This study applies controllable generation using a novel neural-network-based generative model to derive stable boundary stencils for high-order finite-difference schemes. Generative models have been widely used for image generation. They learn the non-linear mapping from a latent distribution to the real data and, given an input noise, generate images of a certain class. Selective manipulation of the input, called controllable generation, then allows change in target attributes, e.g. age or hair color in facial images, while preserving other image features. The generative approach to map a random distribution to stable boundary stencil coefficients is applied in two steps. In the first step, the weights & biases of a neural network (NN) are determined to minimize a cost function comprised of the unstable eigenvalues of the Lyapunov operator for the system matrix. In the second step, the NN weights & biases are fixed and the input is modified to obtain stable stencil coefficients. The efficacy of the approach is demonstrated for various problems.