arXiv:2508. 21571v2 Announce Type: replace Abstract: Physics informed neural networks (PINNs) represent a very popular class of neural solvers for partial differential 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:2605. 03542v2 Announce Type: replace-cross Abstract: The dual norm characterisation of weak solutions of second-order linear elliptic partial differential equations is mathematically natural but computationally intractable: evaluating the $H^{-1}$ norm of the residual requires a supremum over an infinite-dimensional test space.
arXiv:2406. 14340v2 Announce Type: replace-cross Abstract: The standard stochastic gradient descent (SGD) optimization method, as well as adaptive methods such as the Adam optimizer fail to converge if the learning rates do not converge to zero (particularly, in the situation of constant learning rates).
arXiv:2607. 02194v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) have emerged as a promising route to solve partial differential equations, yet they have struggled to reach the precision of classical solvers.
arXiv:2607. 23940v1 Announce Type: new Abstract: Physics-Informed Neural Networks (PINNs) solve differential equations by minimizing the residual of a nonlinear operator over a neural parameterization of the solution.
arXiv:2608. 09494v1 Announce Type: cross Abstract: In this paper we provide Monte Carlo and deep neural network approximations for stochastic representations of solutions to linear elliptic partial differential equations with constant diffusion, drift and killing.
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:2402. 00152v5 Announce Type: replace Abstract: Constructing the architecture of a neural network is a challenging pursuit for the machine learning community, and the dilemma of whether to go deeper or wider remains a persistent question.
arXiv:2606. 16510v1 Announce Type: cross Abstract: This study proposes a Petrov-Galerkin based Variational Physics-Informed Neural Network (VPINN) for efficiently solving two-dimensional singularly perturbed problems (SPPs) with one and two small perturbation parameters.
arXiv:2607. 10200v1 Announce Type: new Abstract: The Neural Tangent Kernel (NTK) is one powerful tool for analyzing the training dynamics of neural networks in the over-parameterized regime.
arXiv:2606. 10562v1 Announce Type: cross Abstract: We propose a new optimization method, the Nystr\"om-enhanced relaxed scalar auxiliary variable method (N-RSAV), which incorporates curvature information into the RSAV framework to accelerate convergence while preserving an unconditional modified energy dissipation law.
arXiv:2606. 19754v1 Announce Type: new Abstract: Partial differential equations (PDEs) play a central role in modeling complex physical, biological, and engineering systems.