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:2607. 20378v1 Announce Type: new Abstract: Physics-informed learning of partial differential equations (PDEs) has been dominated by multilayer perceptrons (MLPs), whose spectral bias and dense parameterization limit both accuracy and interpretability.
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: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.
The paper proposes a unified framework that combines physics‑informed neural networks (PINNs) and finite element methods (FEM) by discretizing functional Gauss–Newton problems using finite families of linear measurements. By interpreting these measurements as test functions, the resulting Gauss–Newton system becomes a Petrov–Galerkin discretization of the linearized functional problem, thereby encompassing pointwise collocation and natural‑gradient approaches as special cases. The framework is specialized to elliptic partial differential equations, yielding weak residual formulations and a hybrid finite‑element–neural architecture that operates on complementary approximation spaces, with numerical experiments confirming its effectiveness.
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:2609.07437v1 Announce Type: cross Abstract: Physics-informed neural networks (PINNs) represent a growing frontier in using artificial intelligence to solve partial differential equations (PDEs)...
The paper presents a theoretical framework for certifying the accuracy of physics‑informed neural networks (PINNs) used to solve partial differential equations. It derives generalization bounds that link the residual loss minimized during training to the actual error in the solution space, showing that if the neural approximation stays within a compact subset, a vanishing residual guarantees convergence to the true solution. Deterministic and probabilistic convergence results are provided, offering explicit error guarantees based on residual, boundary, and initial condition errors.
arXiv:2505. 12430v2 Announce Type: replace Abstract: Recently, innovative adaptations of the Ritz method incorporating deep learning have been developed, known as the Deep Ritz Method.
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: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: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.
arXiv:2601. 07397v2 Announce Type: replace-cross Abstract: In this work, we propose a novel layerwise adaptive construction method for neural network architectures.