Neural Networks with Local Converging Inputs (NNLCI) for Solving Conservation Laws, Part II: 2D Problems (Communications in Computational Physics, 2023)
In our prior work [10], neural networks with local converging inputs(NNLCI) were introduced for solving one-dimensional conservation equations. Twosolutions of a conservation law in a converging sequence, computed from low-costnumerical schemes, and in a local domain of dependence of the space-time location,were used as the input to a neural network in order to predict a high-fidelity solutionat a given space-time location. In the present work, we extend the method to two-dimensional conservation systems and introduce different solution techniques. Numerical results demonstrate the validity and effectiveness of the NNLCI method forapplication to multi-dimensional problems. In spite of low-cost smeared input data,the NNLCI method is capable of accurately predicting shocks, contact discontinuities,and the smooth region of the entire field. The NNLCI method is relatively easy to trainbecause of the use of local solvers. The computing time saving is between one andtwo orders of magnitude compared with the corresponding high-fidelity schemes fortwo-dimensional Riemann problems. The relative efficiency of the NNLCI method isexpected to be substantially greater for problems with higher spatial dimensions orsmooth solutions.