AI & ML
Neural Networks with Local Converging Inputs for Solving the 3D Stokes Equations on Unstructured Meshes: Novel Techniques and Advances
arXiv cs.NA0 views
arXiv:2611.12963v1 Announce Type: cross Abstract: Neural-network surrogates for fluid-flow PDEs usually learn the solution globally, which requires large networks and many high-fidelity training simulations. Neural networks with local converging inputs (NNLCI) instead work locally. They predict the high-fidelity solution at a point from two converging coarse-grid solutions sampled near that point, which keeps the network small, needs little training data, and generalizes across geometries. Here, the two low-fidelity meshes, LF1 and LF2, have on average about 300 and 730 tetrahedra, compared with about 155,000 in the high-fidelity mesh (HF). In this work, we develop an NNLCI framework for three-dimensional Stokes equations on unstructured tetrahedral meshes and introduce two techniques for building local inputs near complex curved boundaries. First, input patches may extend beyond the boundary, with the outside values obtained from a local least-squares fit, so that every high-fidelity point can be predicted, including those next to the wall. Second, the local mesh spacing is estimated from neighboring centroids, which sets the stencil size consistently on unstructured meshes. For pipe flows with one to three bumps, including a circumferentially rotated bump, NNLCI reduces the relative $L^2$ error of the finer low-fidelity solution by $90\%$ to $94\%$, evaluated at all high-fidelity points. Both techniques carry over to other NNLCI problems with smooth solutions on point-cloud discretizations.
Source:https://arxiv.org/abs/2611.12963PreviousA fourth-order dimensionally split exponential time differencing scheme with real and distinct pole rational approximation for non-linear reaction-diffusion systemsNextA constrained mixed variational formulation with Legendre-dual defects for incremental irreversible thermodynamics
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