Nonlinear elliptic homogenization with the parametric Deep Ritz method
Read the original on arXiv Machine Learning →The Flow has not summarised this story yet — read it at arXiv Machine Learning.
The Flow has not summarised this story yet — read it at arXiv Machine Learning.
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:2607. 12570v1 Announce Type: cross Abstract: Multiscale problems are notoriously difficult to tackle using traditional numerical methods, as accurately resolving fine-scale features often requires prohibitively fine discretizations.
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.
arXiv:2603.27936v3 Announce Type: replace-cross Abstract: Nonlinear Partial Differential Equations (PDEs) are ubiquitous in mathematical physics and engineering. Although Physics-Informed Neural Netw...
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.
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.