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Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs
Justin Sirignano, Konstantinos Spiliopoulos, Samuel Cohen
The Deep Galerkin Method (DGM) and Physics Informed Neural Networks (PINNs) have become widely-used methods for solving partial differential equations (PDEs) in the rapidly growing…
Convergence Analysis of Newton's Method for Neural Networks in the Overparameterized Limit
Konstantin Riedl, Konstantinos Spiliopoulos, Justin Sirignano
A convergence analysis is developed for the regularized Newton method for training neural networks (NNs) in the overparameterized limit. As the number of hidden units tends to infi…
Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Control
Samuel N. Cohen, Filippo de Feo, Jackson Hebner +1
We develop deep learning-based approximation methods for fully nonlinear second-order PDEs on separable Hilbert spaces, such as HJB equations for infinite-dimensional control, by p…
Weak Convergence Analysis of Online Neural Actor-Critic Algorithms
Samuel Chun-Hei Lam, Justin Sirignano, Ziheng Wang
We prove that a single-layer neural network trained with the online actor critic algorithm converges in distribution to a random ordinary differential equation (ODE) as the number…
Continuous-time stochastic gradient descent for optimizing over the stationary distribution of stochastic differential equations
Ziheng Wang, Justin Sirignano
We develop a new continuous-time stochastic gradient descent method for optimizing over the stationary distribution of stochastic differential equation (SDE) models. The algorithm…
Global Convergence of the ODE Limit for Online Actor-Critic Algorithms in Reinforcement Learning
Ziheng Wang, Justin Sirignano
Actor-critic algorithms are widely used in reinforcement learning, but are challenging to mathematically analyse due to the online arrival of non-i.i.d. data samples. The distribut…