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.
By William Hofgard, Jingruo Sun, Asaf Cohen
arXiv:2408. 11266v5 Announce Type: replace Abstract: Deep learning is now common across many scientific fields, including the study of partial differential equations.
By Georgios Is. Detorakis
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.
By Justin Sirignano, Konstantinos Spiliopoulos, Samuel Cohen
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.
By Tak Shing Au Yeung, Ka Chun Cheung, Hannah Potgieter, Steven J. Ruuth, Simon See
arXiv:2608. 06595v1 Announce Type: cross Abstract: Neural networks applied to sequential decision-making tasks typically rely on latent representations of environment states.
By Mohamed Ghanem, Bernd Finkbeiner
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.
By Oluwatosin Akande, Gabriel P. Langlois, Akwum Onwunta
arXiv:2606. 27029v1 Announce Type: new Abstract: Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency.
By Harsh Choudhary, Vyacheslav Kungurtsev, Chandan Gupta, Melvin Leok, Georgios Korpas
arXiv:2606. 27029v2 Announce Type: replace Abstract: Hamiltonian Neural Networks (HNNs) integrate physical priors into neural models by learning a system's Hamiltonian, improving generalization and sample efficiency.
By Harsh Choudhary, Vyacheslav Kungurtsev, Chandan Gupta, Melvin Leok, Georgios Korpas
arXiv:2602. 06842v2 Announce Type: replace-cross Abstract: Deep learning-based hybrid iterative methods (DL-HIMs) integrate classical numerical solvers with neural operators, utilizing their complementary spectral biases to accelerate convergence.
By Yuhan Wu, Jan Willem van Beek, Victorita Dolean, Alexander Heinlein
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.
By Hee Jun Yang, Minjung Gim, Yeoneung Kim
arXiv:2602.08515v3 Announce Type: replace-cross
Abstract: This work investigates shallow physics-informed neural networks (PINNs) for solving forward and inverse problems governed by nonlinear partia...
By Muhammad Luthfi Shahab, Imam Mukhlash, Hadi Susanto
arXiv:2606. 19895v1 Announce Type: cross Abstract: The matrices arising from large scale $N$-body problems can be efficiently represented using hierarchical matrices, whose key idea is that the admissible off-diagonal sub-matrices can be well approximated by low-rank matrices across a hierarchy of matrix partitions.
By Jashwanth Reddy Kadaru, Vaishnavi Gujjula