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

Jerome Darbon
Brown University

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.
This is a joint work with Gabriel P. Langlois and Tingwei Meng.

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