# Extensions of the Deep Galerkin Method

arXiv: https://arxiv.org/abs/1912.01455 · HTML: https://mlsanity.com/paper/1912.01455/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Nov 2019.

## Built on by (most cited first, 10 of 10)

- [Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs](https://mlsanity.com/paper/2405.05192/) (2024)
- [Deep Backward and Galerkin Methods for the Finite State Master Equation](https://mlsanity.com/paper/2403.04975/) (2024)
- [Deep learning approximations for non-local nonlinear PDEs with Neumann boundary conditions](https://mlsanity.com/paper/2205.03672/) (2022)
- [Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in Lp-sense](https://mlsanity.com/paper/2409.20431/) (2024)
- [Neural Actor-Critic Methods for Hamilton-Jacobi-Bellman PDEs: Asymptotic Analysis and Numerical Studies](https://mlsanity.com/paper/2507.06428/) (2025)
- [Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control](https://mlsanity.com/paper/2603.19463/) (2026)
- [Net-Zero: A Comparative Study on Neural Network Design for Climate-Economic PDEs Under Uncertainty](https://mlsanity.com/paper/2505.13264/) (2025)
- [DeepPAAC: A New Deep Galerkin Method for Principal-Agent Problems](https://mlsanity.com/paper/2511.04309/) (2025)
- [Randomized Neural Networks for Integro-Differential Equations with Application to Neutron Transport](https://mlsanity.com/paper/2604.13830/) (2026)
- [Deep numerical schemes for systems of Ergodic BSDEs with applications to regime-switching forward utilities](https://mlsanity.com/paper/2606.24271/) (2026)

## Abstract (arXiv)

We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we consider PDEs where the function is constrained to be positive and integrate to unity, as is the case with Fokker-Planck equations. Our approach involves reparameterizing the solution as the exponential of a neural network appropriately normalized to ensure both requirements are satisfied. This then gives rise to nonlinear a partial integro-differential equation (PIDE) where the integral appearing in the equation is handled by a novel application of importance sampling. Secondly, we tackle a number of Hamilton-Jacobi-Bellman (HJB) equations that appear in stochastic optimal control problems. The key contribution is that these equations are approached in their unsimplified primal form which includes an optimization problem as part of the equation. We extend the DGM algorithm to solve for the value function and the optimal control simultaneously by characterizing both as deep neural networks. Training the networks is performed by taking alternating stochastic gradient descent steps for the two functions, a technique inspired by the policy improvement algorithms (PIA).