  We propose reproducing activation functions (RAFs) to improve deep learning
accuracy for various applications ranging from computer vision to scientific
computing. The idea is to employ several basic functions and their learnable
linear combination to construct neuron-wise data-driven activation functions
for each neuron. Armed with RAFs, neural networks (NNs) can reproduce
traditional approximation tools and, therefore, approximate target functions
with a smaller number of parameters than traditional NNs. In NN training, RAFs
can generate neural tangent kernels (NTKs) with a better condition number than
traditional activation functions lessening the spectral bias of deep learning.
As demonstrated by extensive numerical tests, the proposed RAFs can facilitate
the convergence of deep learning optimization for a solution with higher
accuracy than existing deep learning solvers for audio/image/video
reconstruction, PDEs, and eigenvalue problems. With RAFs, the errors of
audio/video reconstruction, PDEs, and eigenvalue problems are decreased by over
14%, 73%, 99%, respectively, compared with baseline, while the performance of
image reconstruction increases by 58%.
