A novel neural network method is developed for solving systems of conservation laws whose solutions may contain abrupt changes of state,including shock waves and contact discontinuities.In conventional approaches,a lo...A novel neural network method is developed for solving systems of conservation laws whose solutions may contain abrupt changes of state,including shock waves and contact discontinuities.In conventional approaches,a low-cost solution patch is usually used as the input to a neural network for predicting the high-fidelity solution patch.With that technique,however,there is no way to distinguish a smeared discontinuity from a smooth solution with large gradient in the input,and the two almost identical inputs correspond to two fundamentally different high-fidelity solution patches in training and predicting.To circumvent this difficulty,we use local patches of two low-cost numerical solutions of the conservation laws in a converging sequence as the input to a neural network.The neural network then makes a correct prediction by identifying whether the solution contains discontinuities or just smooth variations with large gradients,because the former becomes increasingly steep in a converging sequence in the input,and the latter does not.The inputs can be computed from lowcost numerical schemes with coarse resolution,in a local domain of dependence of a space-time location where the prediction is to be made.Despite smeared input solutions,the output provides sharp approximations of solutions containing shock waves and contact discontinuities.The method works effectively not only for regions with discontinuities,but also for smooth regions of the solution.It is efficient to implement,once trained,and has broader applications for different types of differential equations.展开更多
In our prior work[10],neural networks with local converging inputs(NNLCI)were introduced for solving one-dimensional conservation equations.Two solutions of a conservation law in a converging sequence,computed from lo...In our prior work[10],neural networks with local converging inputs(NNLCI)were introduced for solving one-dimensional conservation equations.Two solutions of a conservation law in a converging sequence,computed from low-cost numerical 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 solution at a given space-time location.In the present work,we extend the method to twodimensional conservation systems and introduce different solution techniques.Numerical results demonstrate the validity and effectiveness of the NNLCI method for application 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 train because of the use of local solvers.The computing time saving is between one and two orders of magnitude compared with the corresponding high-fidelity schemes for two-dimensional Riemann problems.The relative efficiency of the NNLCI method is expected to be substantially greater for problems with higher spatial dimensions or smooth solutions.展开更多
基金supported in part by NSF grant DMS-1522585partly sponsored by the Ralph N.Read Endowment of the Georgia Institute of Technology.
文摘A novel neural network method is developed for solving systems of conservation laws whose solutions may contain abrupt changes of state,including shock waves and contact discontinuities.In conventional approaches,a low-cost solution patch is usually used as the input to a neural network for predicting the high-fidelity solution patch.With that technique,however,there is no way to distinguish a smeared discontinuity from a smooth solution with large gradient in the input,and the two almost identical inputs correspond to two fundamentally different high-fidelity solution patches in training and predicting.To circumvent this difficulty,we use local patches of two low-cost numerical solutions of the conservation laws in a converging sequence as the input to a neural network.The neural network then makes a correct prediction by identifying whether the solution contains discontinuities or just smooth variations with large gradients,because the former becomes increasingly steep in a converging sequence in the input,and the latter does not.The inputs can be computed from lowcost numerical schemes with coarse resolution,in a local domain of dependence of a space-time location where the prediction is to be made.Despite smeared input solutions,the output provides sharp approximations of solutions containing shock waves and contact discontinuities.The method works effectively not only for regions with discontinuities,but also for smooth regions of the solution.It is efficient to implement,once trained,and has broader applications for different types of differential equations.
文摘In our prior work[10],neural networks with local converging inputs(NNLCI)were introduced for solving one-dimensional conservation equations.Two solutions of a conservation law in a converging sequence,computed from low-cost numerical 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 solution at a given space-time location.In the present work,we extend the method to twodimensional conservation systems and introduce different solution techniques.Numerical results demonstrate the validity and effectiveness of the NNLCI method for application 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 train because of the use of local solvers.The computing time saving is between one and two orders of magnitude compared with the corresponding high-fidelity schemes for two-dimensional Riemann problems.The relative efficiency of the NNLCI method is expected to be substantially greater for problems with higher spatial dimensions or smooth solutions.
