Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures
Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures
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DOI：
10.1007/s40687-020-00215-6
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发表时间：
2020-07-14
影响因子：
1.2
通讯作者：
Meng, Tingwei
中科院分区：
文献类型：
--
作者：
Darbon, Jerome;Langlois, Gabriel P.;Meng, Tingwei
我们提出了具有初始数据和神经网络结构的Hamilton-Jacobi （HJ）偏微分方程（PDEs）之间新的和原始的数学联系。具体地说，我们证明了一些神经网络对应于HJ PDE解的表示公式，这些解的哈密顿量和初始数据是由神经网络的参数得到的。这些结果不依赖于神经网络的普遍近似性质；相反，我们的结果表明，某些类别的神经网络架构自然地编码了一些HJ pde中包含的物理。我们的结果自然产生了高效的基于神经网络的方法来评估一些高维HJ偏微分方程的解，而不使用网格或数值近似。我们也给出了一些数值结果来解决一些涉及HJ偏微分方程的逆问题。
We propose new and original mathematical connections between Hamilton-Jacobi (HJ) partial differential equations (PDEs) with initial data and neural network architectures. Specifically, we prove that some classes of neural networks correspond to representation formulas of HJ PDE solutions whose Hamiltonians and initial data are obtained from the parameters of the neural networks. These results do not rely on universal approximation properties of neural networks; rather, our results show that some classes of neural network architectures naturally encode the physics contained in some HJ PDEs. Our results naturally yield efficient neural network-based methods for evaluating solutions of some HJ PDEs in high dimension without using grids or numerical approximations. We also present some numerical results for solving some inverse problems involving HJ PDEs using our proposed architectures.