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.

中文翻译：

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通过神经网络架构克服某些Hamilton–Jacobi偏微分方程的维数诅咒

我们提出了汉密尔顿–雅各比（HJ）偏微分方程（PDE）与初始数据和神经网络体系结构之间的新的和原始的数学联系。具体来说，我们证明了某些类型的神经网络对应于HJ PDE解决方案的表示公式，其汉密尔顿函数和初始数据是从神经网络的参数中获得的。这些结果不依赖于神经网络的通用逼近性质。相反，我们的结果表明，某些类型的神经网络体系结构自然会对某些HJ PDE中包含的物理进行编码。我们的结果自然产生了高效的基于神经网络的方法，无需使用网格或数值逼近即可评估高维HJ PDE的解。