We propose novel connections between several neural network architectures and viscosity solutions of some Hamilton–Jacobi (HJ) partial differential equations (PDEs) whose Hamiltonian is convex and only depends on the spatial gradient of the solution. To be specific, we prove that under certain assumptions, the two neural network architectures we proposed represent viscosity solutions to two sets of HJ PDEs with zero error. We also implement our proposed neural network architectures using Tensorflow and provide several examples and illustrations. Note that these neural network representations can avoid curve of dimensionality for certain HJ PDEs, since they do not involve neither grids nor discretization. Our results suggest that efficient dedicated hardware implementation for neural networks can be leveraged to evaluate viscosity solutions of certain HJ PDEs.

我们提出了几种神经网络体系结构和某些哈密顿量为哈密顿量且仅取决于溶液的空间梯度的汉密尔顿-雅各比（HJ）偏微分方程（PDE）的粘性解之间的新颖连接。具体来说，我们证明在某些假设下，我们提出的两种神经网络体系结构代表了两组零误差HJ PDE的粘度解。我们还使用Tensorflow实现了我们提出的神经网络架构，并提供了一些示例和说明。请注意，这些神经网络表示可以避免某些HJ PDE的维数曲线，因为它们既不涉及网格也不涉及离散。