The paper introduces a deep learning algorithm, called the Deep Galerkin Method (DGM), to solve high-dimensional partial differential equations (PDEs). The DGM approximates the solution of a PDE using a deep neural network that is trained to satisfy the differential operator, initial condition, and boundary conditions. Unlike traditional methods that require meshing, the DGM is meshfree, making it feasible for high-dimensional problems. The algorithm is tested on various high-dimensional PDEs, including free boundary PDEs, Hamilton-Jacobi-Bellman PDEs, and Burgers' equation. The authors also prove a theorem showing the approximation power of neural networks for a class of quasilinear parabolic PDEs. The DGM is particularly useful for solving high-dimensional PDEs in finance, such as pricing American options, where the number of dimensions can be dozens or even hundreds. The paper provides detailed implementation details, including the neural network architecture, hyperparameters, and computational approach, and demonstrates the effectiveness of the DGM through numerical experiments.The paper introduces a deep learning algorithm, called the Deep Galerkin Method (DGM), to solve high-dimensional partial differential equations (PDEs). The DGM approximates the solution of a PDE using a deep neural network that is trained to satisfy the differential operator, initial condition, and boundary conditions. Unlike traditional methods that require meshing, the DGM is meshfree, making it feasible for high-dimensional problems. The algorithm is tested on various high-dimensional PDEs, including free boundary PDEs, Hamilton-Jacobi-Bellman PDEs, and Burgers' equation. The authors also prove a theorem showing the approximation power of neural networks for a class of quasilinear parabolic PDEs. The DGM is particularly useful for solving high-dimensional PDEs in finance, such as pricing American options, where the number of dimensions can be dozens or even hundreds. The paper provides detailed implementation details, including the neural network architecture, hyperparameters, and computational approach, and demonstrates the effectiveness of the DGM through numerical experiments.