用可学习频率的切比雪夫函数提升神经网络函数逼近精度
Chebyshev Feature Neural Network for Accurate Function Approximation
- 首层使用可学习频率的切比雪夫函数，覆盖广泛频率范围
- 多阶段训练下实现机器精度的函数逼近，支持20维输入
- 适合高精度科学计算与复杂函数建模任务
我们提出一种新型深度神经网络（DNN）架构——切比雪夫特征神经网络（CFNN），可实现接近机器精度的函数逼近。该结构在第一层采用可学习频率的切比雪夫函数，后续为标准全连接层。切比雪夫层的频率参数初始化为指数分布，以覆盖广泛频率范围。结合多阶段训练策略，实验表明该方法可在训练过程中达到机器精度。通过一系列数值实验验证了该方法在20维以下问题上的有效性与可扩展性。
原文摘要 · Abstract (English)
We present a new Deep Neural Network (DNN) architecture capable of approximating functions up to machine accuracy. Termed Chebyshev Feature Neural Network (CFNN), the new structure employs Chebyshev functions with learnable frequencies as the first hidden layer, followed by the standard fully connected hidden layers. The learnable frequencies of the Chebyshev layer are initialized with exponential distributions to cover a wide range of frequencies. Combined with a multi-stage training strategy, we demonstrate that this CFNN structure can achieve machine accuracy during training. A comprehensive set of numerical examples for dimensions up to $20$ are provided to demonstrate the effectiveness and scalability of the method.
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