首个可同时处理时域与频域的算子网络，误差低于1e-7。
Radial Basis Operator Networks
- 基于径向基函数设计，支持复数输入，实现时空联合建模。
- 在分布内和分布外数据上均达小于1×10⁻⁷的相对误差。
- 适用于跨函数类泛化，适合科学计算中的复杂场建模。
算子网络旨在逼近非线性算子，即在无限维空间（如函数空间）间建立映射。这类网络在机器学习中日益重要，尤其在科学计算领域。其优势在于能处理气候模拟或流体动力学中常见的离散连续场数据（如温度分布或速度场）。本文提出径向基算子网络（RBON），是首个可接受复数输入、在时间域与频率域同步学习算子的架构。尽管结构仅含单层隐藏层，但其在多个基准测试中对分布内和分布外（OOD）数据的$L^2$相对测试误差均低于$1\times 10^{-7}$。此外，该网络在来自训练数据完全不同函数类的分布外数据上仍保持低误差，展现出强大泛化能力。
原文摘要 · Abstract (English)
Operator networks are designed to approximate nonlinear operators, which provide mappings between infinite-dimensional spaces such as function spaces. These networks are playing an increasingly important role in machine learning, with their most notable contributions in the field of scientific computing. Their significance stems from their ability to handle the type of data often encountered in scientific applications. For instance, in climate modeling or fluid dynamics, input data typically consists of discretized continuous fields (like temperature distributions or velocity fields). We introduce the radial basis operator network (RBON), which represents a significant advancement as the first operator network capable of learning an operator in both the time domain and frequency domain when adjusted to accept complex-valued inputs. Despite the small, single hidden-layer structure, the RBON boasts small $L^2$ relative test error for both in- and out-of-distribution data (OOD) of less than $1\times 10^{-7}$ in some benchmark cases. Moreover, the RBON maintains small error on OOD data from entirely different function classes from the training data.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。