RBF-PINN：Physics-Informed Neural Networks中的非Fourier位置嵌入
RBF-PINN: Non-Fourier Positional Embedding in Physics-Informed Neural Networks
AI总结：
针对PINNs中基于Fourier特征映射的局限性，提出使用条件正定的Radial Basis Function作为非Fourier位置嵌入，该方法可无缝集成至坐标输入网络，并在多种正逆问题中验证了有效性。
摘要：
尽管许多最近的Physics-Informed Neural Networks (PINNs)变体在求解偏微分方程方面取得了相当大的成功，但源自更广泛的Neural Representations研究的特征映射的经验优势在很大程度上被忽略了。我们强调了广泛使用的基于Fourier的特征映射在某些情况下的局限性，并建议使用条件正定的Radial Basis Function。经验结果表明了我们的方法在各种正向和反向问题案例中的有效性。我们的方法可以无缝集成到基于坐标的输入神经网络中，并为更广泛的PINNs研究领域做出贡献。
英文摘要：
While many recent Physics-Informed Neural Networks (PINNs) variants have had considerable success in solving Partial Differential Equations, the empirical benefits of feature mapping drawn from the broader Neural Representations research have been largely overlooked. We highlight the limitations of widely used Fourier-based feature mapping in certain situations and suggest the use of the conditionally positive definite Radial Basis Function. The empirical findings demonstrate the effectiveness of our approach across a variety of forward and inverse problem cases. Our method can be seamlessly integrated into coordinate-based input neural networks and contribute to the wider field of PINNs research.
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