arXiv Machine Learning
Sep 15

Physics-Constrained Neural Surrogate for Domain Growth Prediction in Systems with Conserved Kinetics

The paper introduces a physics-constrained neural network surrogate that learns the microstructural evolution of binary mixtures governed by the Cahn‑Hilliard equation. By imposing conservation of the order parameter as a hard constraint on the network output, the model accurately predicts long‑time phase‑separation dynamics for both critical and off‑critical mixtures, maintaining mixture composition and matching the Lifshitz‑Slyozov domain‑growth law. A variant that enforces conservation only through a penalty term drifts from the initial composition and loses predictive accuracy over long rollouts, underscoring the necessity of the hard constraint for stability.

By Vijay Yadav, Pallvi Pandey, Madhu Priya, Manish Dev Shrimali, Prabhat K. Jaiswal
arXiv Machine Learning
Jul 23

PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs

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.

By Amirhossein Sadr, Nima Soltani, Vahideh Moghtadaiee, Aida Pakniyat, Dara Rahmati, Saeid Gorgin
arXiv Machine Learning
Sep 18

Beyond PINNs: A Unified Gauss--Newton and Petrov--Galerkin Framework for Neural and Hybrid PDE Solvers

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.

By Nilo Schwencke, Roland Maier