论文标题
在某些可以代表某些高维汉密尔顿的粘度解决方案的神经网络体系结构上
On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton--Jacobi partial differential equations
论文作者
论文摘要
我们提出了一些汉密尔顿（HJ）部分微分方程（PDE）的几个神经网络体系结构与粘度解决方案之间的新连接，其汉密尔顿是凸的，仅取决于溶液的空间梯度。具体而言，我们证明在某些假设下，我们提出的两个神经网络体系结构代表了两组HJ PDES的粘度解决方案，其误差为零。我们还使用TensorFlow实施了建议的神经网络架构，并提供了几个示例和插图。请注意，这些神经网络表示可以避免某些HJ PDE的维度曲线，因为它们既不涉及网格也不涉及离散化。我们的结果表明，可以利用有效的神经网络的专用硬件实现来评估某些HJ PDE的粘度解决方案。
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.
