We demonstrate that neural networks can be FLOP-efficient integrators of one-dimensional oscillatory integrands. We train a feed-forward neural network to compute integrals of highly oscillatory 1D functions. The training set is a parametric combination of functions with varying characters and oscillatory behavior degrees. Numerical examples show that these networks are FLOP-efficient for sufficiently oscillatory integrands with an average FLOP gain of 1000 FLOPs. The network calculates oscillatory integrals better than traditional quadrature methods under the same computational budget or number of floating point operations. We find that feed-forward networks of 5 hidden layers are satisfactory for a relative accuracy of 0.001. The computational burden of inference of the neural network is relatively small, even compared to inner-product pattern quadrature rules. We postulate that our result follows from learning latent patterns in the oscillatory integrands that are otherwise opaque to traditional numerical integrators.

翻译：我们证明神经网络可作为一维振荡被积函数的FLOP高效积分器。通过训练前馈神经网络计算高度振荡一维函数的积分，训练集由具有不同特性与振荡行为程度的函数参数化组合构成。数值实验表明，对于振荡性充分的被积函数，这些网络可实现FLOP效率提升，平均FLOP增益达1000次。在相同计算预算或浮点运算次数下，神经网络对振荡积分的计算效果优于传统求积方法。研究发现，五层隐藏层的前馈网络即可实现0.001的相对精度。即使与内积模式求积规则相比，神经网络推理的计算负担也相对较小。我们推测该结果源于网络能够学习传统数值积分器无法辨别的振荡被积函数中的潜在模式。