Feature Mapping in Physics-Informed Neural Networks (PINNs)
<|reference_start|>Feature Mapping in Physics-Informed Neural Networks (PINNs): In this paper, the training dynamics of PINNs with a feature mapping layer via the limiting Conjugate Kernel and Neural Tangent Kernel is investigated, shedding light on the convergence of PINNs; Although the commonly used Fourier-based feature mapping has achieved great success, we show its inadequacy in some physics scenarios. Via these two scopes, we propose conditionally positive definite Radial Basis Function as a better alternative. Lastly, we explore the feature mapping numerically in a wide neural networks. Our empirical results reveal the efficacy of our method in diverse forward and inverse problem sets. Composing feature functions is found to be a practical way to address the expressivity and generalisability trade-off, viz., tuning the bandwidth of the kernels and the surjectivity of the feature mapping function. This simple technique can be implemented for coordinate inputs and benefits the broader PINNs research.<|reference_end|>
@article{zeng2024feature,
    title={Feature Mapping in Physics-Informed Neural Networks (PINNs)},
    author={Chengxi Zeng, Tilo Burghardt, Alberto M Gambaruto},
    journal={arXiv preprint arXiv:2402.06955},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.06955},
    primaryClass={cs.LG cs.AI cs.CE}
}