arXiv:2605. 28983v2 Announce Type: replace-cross Abstract: In this paper, training a neural network is identified, exactly, as a search through Hamilton--Jacobi initial-value problems: each gradient step selects the initial data of a viscous Hamilton--Jacobi equation whose Hopf--Cole propagator best fits the observations; at inference, the input is the spatial point at which that solution is evaluated and the initial condition is already encoded in the weights.
By Jose Marie Antonio Mi\~noza, Erika Fille T. Legara, Christopher P. Monterola
arXiv:2607. 19167v1 Announce Type: cross Abstract: Motivated by the numerical computation of the Mean Escape Time (MET) $\tau:\Omega\to\mathbb{R}$ of a stochastic process from a bounded domain $\Omega\subseteq\mathbb{R}^d$, we study elliptic Dirichlet boundary value problems (BVPs) using boundary-enforced Physics-Informed Neural Networks (PINNs), in which the Dirichlet condition is imposed exactly by multiplying the network output with a predefined distance-to-boundary approximation $\rho$.
By Nathanael Tepakbong, Jun Fan, Xiang Zhou, Ding-Xuan Zhou
arXiv:2607. 15702v2 Announce Type: replace-cross Abstract: We develop a non-asymptotic approximation, sampling, and finite-iteration optimization theory for variational physics-informed approximation of uniformly monotone nonlinear multiscale elliptic equations.
By Ronald Katende
arXiv:2610.27193v1 Announce Type: cross
Abstract: Physics-informed neural networks (PINNs) and other residual-trained solvers judge a candidate solution by the size of its sampled PDE residual. For f...
By Raul Jimenez, Pablo Tejerina-P\'erez, Leonid Sarieddine, Pedro Taranc\'on-\'Alvarez, Marisol Traforetti, Pau Sol\'e-Vilar\'o, Oleksii Ignatenko, Nazar Tkach, Pavlos Protopapas
arXiv:2609. 03626v1 Announce Type: cross Abstract: Rigorous results show that feedforward neural networks can overcome the curse of dimensionality in the numerical approximation of high-dimensional partial differential equations (PDEs), but comparatively little is known about residual neural networks (ResNets) in the nonlinear PDE setting.
By Ilkhom Mukhammadiev, Diyora Salimova
The paper introduces a certified continuation framework for computing and training deep equilibrium networks (DEQs). It uses compact input homotopy and a rounded Newton tracker for inference, and augments local-plus-low-rank recurrence with programmable dormant bilinear rank‑one channels for training. The approach guarantees polynomial‑time bit complexity, with certified bounds on inference and training error budgets.
By Alex Borisevich