The field of partial differential equations (PDEs) is experiencing significant advancements with the integration of neural networks and novel numerical methods. Recent developments have focused on improving the accuracy and efficiency of solving PDEs, particularly in complex domains and with highly oscillatory solutions. Graph neural networks and physics-informed neural networks (PINNs) have emerged as powerful tools for solving PDEs, offering improved performance over traditional numerical methods. Additionally, new numerical methods such as the finite expression method and the B-spline collocation method have been proposed to tackle challenging PDE problems. These innovations have far-reaching implications for various fields, including biophysics, materials science, and energy storage. Noteworthy papers include: Graph Neural Regularizers for PDE Inverse Problems, which introduces a framework for solving ill-posed inverse problems using graph neural networks; PINN Balls, which proposes a scalable and accurate method for training PINNs; and LieSolver, which leverages Lie symmetries to efficiently solve initial-boundary value problems. These advancements demonstrate the rapid progress being made in the field and highlight the potential for future breakthroughs.
Advances in Solving Partial Differential Equations with Neural Networks and Novel Numerical Methods
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Graph Neural Regularizers for PDE Inverse Problems
PINN Balls: Scaling Second-Order Methods for PINNs with Domain Decomposition and Adaptive Sampling
A Variational Framework for the Algorithmic Complexity of PDE Solutions
A Rapid Physics-Informed Machine Learning Framework Based on Extreme Learning Machine for Inverse Stefan Problems
Error Estimates for Sparse Tensor Products of B-spline Approximation Spaces
Multi-Scale Finite Expression Method for PDEs with Oscillatory Solutions on Complex Domains
Surface layers and linearized water waves: a boundary integral equation framework
Graph Neural Network Assisted Genetic Algorithm for Structural Dynamic Response and Parameter Optimization
Rational Approximation via p-AAA on Scattered Data Sets
A Finite Element framework for bulk-surface coupled PDEs to solve moving boundary problems in biophysics
The B-spline collocation method for solving Cauchy singular integral equations with piecewise Holder continuous coefficients
Graph Network-based Structural Simulator: Graph Neural Networks for Structural Dynamics
Fourier Neural Operators for Two-Phase, 2D Mold-Filling Problems Related to Metal Casting
LieSolver: A PDE-constrained solver for IBVPs using Lie symmetries
Meshless solutions of PDE inverse problems on irregular geometries
Incorporating Local H\"older Regularity into PINNs for Solving Elliptic PDEs
The evolving surface morphochemical reaction-diffusion system for battery modeling