论文标题
光谱神经操作员
Spectral Neural Operators
论文作者
论文摘要
在科学计算中，大量应用需要在Banach空间之间进行映射的近似值。最近引入的傅立叶神经操作员（FNO）和深层操作员网络（DeepOnet）可以提供此功能。对于这两个神经操作员，在给定的网格（FNO的均匀）上对输入函数进行采样，并且输出函数通过神经网络参数化。我们认为，这种参数化导致1）难以分析的不透明输出，而2）在FNO中由异叠误差引起的系统偏见。本文提倡的替代方法是将Chebyshev和Fourier系列用于域和代码域。所得的频谱神经操作员（SNO）具有透明的输出，从不混杂，并且可能包括许多功能上的精确（无损）操作。该功能基于光谱方法的快速发达和稳定的算法。实施仅需要标准数值线性代数。我们的基准表明，对于许多运营商而言，SNO优于FNO和DEADONET。
A plentitude of applications in scientific computing requires the approximation of mappings between Banach spaces. Recently introduced Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet) can provide this functionality. For both of these neural operators, the input function is sampled on a given grid (uniform for FNO), and the output function is parametrized by a neural network. We argue that this parametrization leads to 1) opaque output that is hard to analyze and 2) systematic bias caused by aliasing errors in the case of FNO. The alternative, advocated in this article, is to use Chebyshev and Fourier series for both domain and codomain. The resulting Spectral Neural Operator (SNO) has transparent output, never suffers from aliasing, and may include many exact (lossless) operations on functions. The functionality is based on well-developed fast, and stable algorithms from spectral methods. The implementation requires only standard numerical linear algebra. Our benchmarks show that for many operators, SNO is superior to FNO and DeepONet.
