The paper proves that the deep Galerkin method (DGM) converges when applied to Hamilton‑Jacobi‑Bellman equations derived from finite‑state mean field control problems. By showing that the DGM loss can be driven arbitrarily low under sufficient regularity of the value function, and that a vanishing loss forces uniform convergence of the neural network approximators to the true value function on the simplex, the authors establish both existence and convergence results for the DGM. Numerical experiments further illustrate the method’s ability to handle high‑dimensional HJB equations.
arXiv:2607. 24726v1 Announce Type: new Abstract: 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 field of scientific machine learning.
arXiv:2511. 04309v3 Announce Type: replace-cross Abstract: We consider numerical resolution of principal-agent (PA) problems in continuous time.
arXiv:2311. 15365v3 Announce Type: replace Abstract: We study an idealized training process for deep neural networks in a continuous-depth, mean-field model in which each layer is parameterized by a probability measure on a Euclidean parameter space.
arXiv:2408. 11266v5 Announce Type: replace Abstract: Deep learning is now common across many scientific fields, including the study of partial differential equations.
arXiv:2607. 02003v1 Announce Type: cross Abstract: Although neural networks are remarkably effective, their underlying optimization principles remain theoretically elusive, often characterized by non-convex landscapes and stochastic heuristics.
The paper introduces residual-based loss functions derived from Discontinuous Petrov Galerkin (DPG) discretizations for training neural networks to learn parameter-to-solution maps of PDEs. It focuses on rigorous accuracy certification and demonstrates the approach on an elliptic PDE, showing that DPG-based losses outperform simple least-squares losses, especially for high-contrast diffusion problems. The concepts are applicable to any problem with a stable DPG formulation.
arXiv:2605. 07116v2 Announce Type: replace Abstract: We analyze a neural semi-discrete method for high-dimensional first-order Hamilton-Jacobi-Bellman (HJB) equations with known or learned dynamics.
The paper introduces a mesh‑free policy iteration framework that blends classical dynamic programming with physics‑informed neural networks (PINNs) to solve high‑dimensional, nonconvex Hamilton–Jacobi–Isaacs (HJI) equations. The method alternates between solving linear second‑order PDEs under fixed feedback policies and updating controls via pointwise minimax optimization using automatic differentiation. The authors prove local uniform convergence of the value function iterates to the unique viscosity solution under standard Lipschitz and uniform ellipticity assumptions, and demonstrate the approach’s accuracy and scalability in two‑, five‑, and ten‑dimensional stochastic games, outperforming direct PINN solvers.
arXiv:2606. 24999v1 Announce Type: new Abstract: High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging.
arXiv:2404. 05185v4 Announce Type: replace-cross Abstract: This paper deals with a class of neural SDEs and studies the limiting behavior of the associated sampled optimal control problems as the sample size grows to infinity.
The paper explores neural network solvers for infinity and p‑Laplace problems, employing Physics‑Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONets). It addresses computational challenges for large p values (2 to 1000) across 2D and 3D domains, showing advantages over traditional mesh‑based solvers, especially in three dimensions. The authors provide conditional convergence results for PINNs, a universal approximation theorem for DeepONet on the parametric p‑Poisson problem, and validate their methods with numerical experiments comparing performance to conventional approaches.
The paper proposes a deep learning framework that learns priors for inverse problems by exploiting the relationship between proximal operators and Hamilton–Jacobi partial differential equations. Unlike existing methods that require inverting the prior after training, this approach learns the prior directly, enabling efficient evaluation in a single forward pass. Numerical experiments demonstrate the method’s effectiveness in dimensions up to 64.