arXiv Machine Learning July 22, 2026 By Nathanael Tepakbong, Jun Fan, Xiang Zhou, Ding-Xuan Zhou

Boundary-Adapted PINNs for Elliptic Dirichlet Problems: $H^2(\Omega)$ A Priori Error Bounds with Application to Mean Escape Time Computation

Read the original on arXiv Machine Learning →

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 $\rho$.

Summary generated by The Flow from the publisher's feed. The full article lives at arXiv Machine Learning.

arXiv Machine Learning

Jun 2

Taming the Loss Landscape of PINNs with Noisy Feynman-Kac Supervision: Operator Preconditioning and Non-Asymptotic Error Bounds

arXiv:2606. 00643v1 Announce Type: cross Abstract: Physics-Informed Neural Networks (PINNs) often train slowly or fail to converge on challenging partial differential equations (PDEs), a behavior recently linked to severely ill-conditioned loss landscapes inherited from the underlying differential operator.

By Nathanael Tepakbong, Hanyu Hu, Chengyu Liu, Xiang Zhou

arXiv Machine Learning

Jun 26

Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks

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.

By Diego Marcondes

arXiv Machine Learning

Aug 4

Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation

arXiv:2605. 08170v2 Announce Type: replace Abstract: Neural operators have emerged as a powerful tool for learning mappings between infinite-dimensional function spaces.

By Nicole Hao

arXiv Machine Learning

Jul 3

An Optimisation Framework for the Well-Conditioned Training of Physics-Informed Neural Networks

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.

By Joseph Webb, Sadok Jerad, Coralia Cartis

arXiv Machine Learning

Jun 11

Reliable Error Estimation for PINNs: Lower and Upper A Posteriori Bounds

arXiv:2606. 12050v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) combine machine learning with physical laws to solve differential equations.

By Ismail Huseynov, Arzu Ahmadova, Agamirza Bashirov

arXiv Machine Learning

Jul 27

Approximation and learning of anisotropic and mixed smooth functions by deep ReLU neural networks

arXiv:2605. 31152v2 Announce Type: replace-cross Abstract: This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions.

By Yunfei Yang, Jun Fan

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Taming the Loss Landscape of PINNs with Noisy Feynman-Kac Supervision: Operator Preconditioning and Non-Asymptotic Error Bounds

Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks

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