We investigate the concept of Best Approximation for Feedforward Neural Networks (FNN) and explore their convergence properties through the lens of Random Projection (RPNNs). RPNNs have predetermined and fixed, once and for all, internal weights and biases, offering computational efficiency. We demonstrate that there exists a choice of external weights, for any family of such RPNNs, with non-polynomial infinitely differentiable activation functions, that exhibit an exponential convergence rate when approximating any infinitely differentiable function. For illustration purposes, we test the proposed RPNN-based function approximation, with parsimoniously chosen basis functions, across five benchmark function approximation problems. Results show that RPNNs achieve comparable performance to established methods such as Legendre Polynomials, highlighting their potential for efficient and accurate function approximation.

翻译：我们研究了前馈神经网络（FNN）的最佳逼近概念，并通过随机投影神经网络（RPNN）的视角探索其收敛性质。RPNN具有预先设定且固定不变的内部权重和偏置，具有计算高效性。我们证明，对于任意一类具有非多项式无穷可微激活函数的RPNN，当逼近任意无穷可微函数时，存在一组外部权重选择可实现指数级收敛速度。为说明该结论，我们基于所提出的RPNN函数逼近方法，在五个基准函数逼近问题上进行了测试，并采用简约选取的基函数。结果表明，RPNN能够达到与勒让德多项式等经典方法相当的逼近性能，凸显了其在高效且精确的函数逼近中的潜力。