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)"  .
High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator , initial condition, and boundary conditions. We prove that the neural network converges to the solution of the partial differential equation as the number of hidden units increases. Our algorithm is meshfree , which is key since meshes become infeasible in higher dimensions. Instead of forming a mesh, the neural network is trained on batches of randomly sampled time and space points . We implement the approach for American options (a type of free-boundary PDE which is widely used in finance) in up to 200 dimensions. We call the algorithm a "Deep Galerkin Method (DGM)" since it is similar in spirit to Galerkin methods, with the solution approximated by a neural network instead of a linear combination of basis functions .
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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
to solve high-dimensional PDEs by approximating the solution
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,
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on random batches of spatial points
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, initial condition,
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The algorithm is mesh-less
We prove that the neural network converges to the solution of the partial differential equation as the number of hidden units increases. Our algorithm is meshfree
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sequences of spatial points are randomly sampled
the neural network is trained on batches of randomly sampled time and space points
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100
200
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since it is similar in spirit to Galerkin methods, with the solution approximated by a neural network instead of a linear combination of basis functions