High-dimensional PDEs have been a longstanding computational challenge. We propose a deep learning algorithm similar in spirit to Galerkin methods, using a deep neural network instead of linear combinations of basis functions. The PDE is approximated with a deep neural network, which is trained on random batches of spatial points to satisfy the differential operator and boundary conditions. The algorithm is mesh-less, which is key since meshes become infeasible in higher dimensions. Instead of forming a mesh, sequences of spatial points are randomly sampled. We implement the approach for American options (a type of free-boundary PDE which is widely used in finance) in up to 100 dimensions. We call the algorithm a "Deep Galerkin Method (DGM)".
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