This paper introduces the Deep Galerkin Method (DGM), a deep learning algorithm for solving high-dimensional partial differential equations (PDEs). The method approximates the solution using a deep neural network trained to satisfy the differential operator, initial condition, and boundary conditions. Unlike traditional mesh-based methods, DGM is mesh-free, making it suitable for high-dimensional problems where meshes become infeasible. The algorithm is tested on various high-dimensional PDEs, including free boundary PDEs, Hamilton-Jacobi-Bellman PDEs, and Burgers' equation. It is shown that DGM can accurately solve these PDEs in up to 200 dimensions. The method is also proven to have approximation power for a class of quasilinear parabolic PDEs. The algorithm uses stochastic gradient descent to train the neural network on randomly sampled spatial points, avoiding the need for a mesh. A Monte Carlo method is introduced to efficiently compute second derivatives, which are essential for the algorithm. The DGM algorithm is implemented and tested on a class of high-dimensional free boundary PDEs, demonstrating its effectiveness in solving complex PDEs in high dimensions. The method is also shown to handle cases where no semi-analytic solution exists, as the error bounds for approximate solutions can be calculated. The algorithm is efficient and scalable, with the ability to process large numbers of spatial points sequentially without harming the convergence rate. The paper also discusses the implementation details of the algorithm, including the neural network architecture, hyperparameters, and computational approach. The results show that DGM can accurately approximate the solution to PDEs in high dimensions, making it a promising approach for solving complex PDEs in various scientific and engineering applications.This paper introduces the Deep Galerkin Method (DGM), a deep learning algorithm for solving high-dimensional partial differential equations (PDEs). The method approximates the solution using a deep neural network trained to satisfy the differential operator, initial condition, and boundary conditions. Unlike traditional mesh-based methods, DGM is mesh-free, making it suitable for high-dimensional problems where meshes become infeasible. The algorithm is tested on various high-dimensional PDEs, including free boundary PDEs, Hamilton-Jacobi-Bellman PDEs, and Burgers' equation. It is shown that DGM can accurately solve these PDEs in up to 200 dimensions. The method is also proven to have approximation power for a class of quasilinear parabolic PDEs. The algorithm uses stochastic gradient descent to train the neural network on randomly sampled spatial points, avoiding the need for a mesh. A Monte Carlo method is introduced to efficiently compute second derivatives, which are essential for the algorithm. The DGM algorithm is implemented and tested on a class of high-dimensional free boundary PDEs, demonstrating its effectiveness in solving complex PDEs in high dimensions. The method is also shown to handle cases where no semi-analytic solution exists, as the error bounds for approximate solutions can be calculated. The algorithm is efficient and scalable, with the ability to process large numbers of spatial points sequentially without harming the convergence rate. The paper also discusses the implementation details of the algorithm, including the neural network architecture, hyperparameters, and computational approach. The results show that DGM can accurately approximate the solution to PDEs in high dimensions, making it a promising approach for solving complex PDEs in various scientific and engineering applications.