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We propose new and original mathematical connections between Hamilton-Jacobi(HJ) partial differential equations (PDEs) with initial data and neural networkarchitectures. Specifically, we prove that some classes of neural networkscorrespond to representation formulas of HJ PDE solutions whose Hamiltoniansand initial data are obtained from the parameters of the neural networks. Theseresults do not rely on universal approximation properties of neural networks;rather, our results show that some classes of neural network architecturesnaturally encode the physics contained in some HJ PDEs. Our results naturallyyield efficient neural network-based methods for evaluating solutions of someHJ PDEs in high dimension without using grids or numerical approximations. Wealso present some numerical results for solving some inverse problems involvingHJ PDEs using our proposed architectures.

Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures