Petrov-Galerkin Variational Physics-Informed Neural Network Framework for Two-Dimensional Singularly Perturbed Problems
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. The approach employs neural networks to construct the trial solution space, while tensor-product hat functions are adopted as test functions to enforce the variational form. To accurately resolve of sharp boundary layers, the variational form is implemented using a Petrov-Galerkin formulation. Dirichlet boundary conditions are
The rapid advancement in neural network architectures and computational capabilities allows for more sophisticated applications of AI in scientific computing, driven by the ongoing demand for efficient solutions to complex mathematical problems.
This development indicates continued progress in using AI to solve computationally intensive scientific and engineering problems, potentially accelerating research and development in fields reliant on differential equations.
The use of Petrov-Galerkin methods within Physics-Informed Neural Networks (PINNs) offers a more robust and accurate way to handle challenging 'singularly perturbed problems,' which are common in many scientific and engineering domains.
- · AI/ML researchers and developers
- · Scientific computing software companies
- · Engineering and physics research sectors
- · High-performance computing providers
- · Traditional numerical methods that are less accurate or efficient in these probl
Improved accuracy and efficiency in solving complex differential equations, especially those with sharp boundary layers.
Faster simulation and design cycles in fields like fluid dynamics, materials science, and climate modeling due to better problem-solving tools.
Reduced development costs and accelerated innovation in industries that heavily rely on computationally expensive simulations.
This signal links to a primary source. Continuum Brief monitors and indexes it as part of the live intelligence stream — we do not republish source content.
Read at arXiv cs.LG