Advances in Physics-Informed Neural Networks for PDE Solving

The field of physics-informed neural networks (PINNs) for solving partial differential equations (PDEs) is rapidly advancing, with a focus on improving accuracy, stability, and efficiency. Recent developments have introduced novel architectures and techniques, such as integral regularization, domain decomposition, and implicit neural differential models, to tackle challenging problems in PDE solving. These innovations have shown promising results in capturing long-time behaviors, generalizing to new geometries, and reducing computational costs. Noteworthy papers in this area include those that propose operator learning with domain decomposition for geometry generalization and implicit neural differential models for spatiotemporal dynamics. Overall, the field is moving towards more robust, accurate, and efficient solutions for PDE solving, with potential applications in various fields such as computational mechanics and scientific machine learning.

Sources

Generative Latent Neural PDE Solver using Flow Matching

Integral regularization PINNs for evolution equations

Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solving

Hard-constraining Neumann boundary conditions in physics-informed neural networks via Fourier feature embeddings

Representing Flow Fields with Divergence-Free Kernels for Reconstruction

Implicit Neural Differential Model for Spatiotemporal Dynamics

A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries

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