Approximating Nonlocal Periodic Operators with Neural Networks: A PDE Perspective with Applications to the Critical SQG Equation
This paper introduces a novel PDE-based approach to approximate nonlocal periodic operators using neural networks, which traditionally lack periodic boundary conditions.
Error Analysis and Numerical Validation of the Orthogonal Greedy Algorithm for Solving Indefinite Elliptic Problems Using Shallow Neural Networks
This research paper presents a rigorous error analysis of using shallow neural networks, trained with the Orthogonal Greedy Algorithm (OGA), to solve indefinite elliptic problems, demonstrating the method's effectiveness and superior performance compared to traditional finite element methods.
Adaptive Finite Element Interpolated Neural Networks Analysis
Neural networks are used to approximate partial differential equations with local phenomena, introducing adaptive finite element interpolated neural networks.
Adversarial Adaptive Sampling: Unifying PINN and Optimal Transport for PDE Approximation at ICLR 2024
提案されたAdversarial Adaptive Sampling（AAS）アプローチは、PDEのニューラルネットワーク近似においてPINNと最適輸送を統合しました。
Overcoming the Curse of Dimensionality in Neural Network Approximation of PDEs
Deep neural networks can overcome the curse of dimensionality in approximating solutions of certain nonlinear PDEs.
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