Neural Processes (NPs) are meta-learning models that learn to map sets of observations to approximations of the corresponding posterior predictive distributions. By accommodating variable-sized, unstructured collections of observations and enabling probabilistic predictions at arbitrary query points, NPs provide a flexible framework for modeling functions over continuous domains. Since their introduction, numerous variants have emerged; however, early formulations shared a fundamental limitation: they compressed the observed data into finite-dimensional global representations via aggregation operations such as mean pooling. This strategy induces an intrinsic mismatch with the infinite-dimensional nature of the stochastic processes that NPs intend to model. Convolutional conditional neural processes (ConvCNPs) address this limitation by constructing infinite-dimensional functional embeddings processed through convolutional neural networks (CNNs) to enforce translation equivariance. Yet CNNs with local spatial kernels struggle to capture long-range dependencies without resorting to large kernels, which impose significant computational costs. To overcome this limitation, we propose spectral ConvCNPs (SConvCNPs), which perform global convolution in the frequency domain. Inspired by Fourier neural operators (FNOs) for learning solution operators of partial differential equations (PDEs), our approach directly parameterizes convolution kernels in the frequency domain, leveraging the relatively compact yet global Fourier representation of many natural signals. We validate the effectiveness of SConvCNPs on both synthetic and real-world datasets, demonstrating how ideas from operator learning can advance the capabilities of NPs.

翻译：神经过程（NPs）是一种元学习模型，其学习将观测集映射至对应后验预测分布的近似。通过容纳可变尺寸、非结构化的观测集合，并支持在任意查询点进行概率预测，NPs为连续域上的函数建模提供了一个灵活框架。自其提出以来，已涌现出众多变体；然而，早期版本存在一个根本性局限：它们通过均值池化等聚合操作将观测数据压缩为有限维的全局表示。这一策略与NPs旨在建模的随机过程的无限维本质存在内在不匹配。卷积条件神经过程（ConvCNPs）通过构建无限维函数嵌入，并利用卷积神经网络（CNNs）进行处理以强制平移等变性，从而解决了这一局限。然而，采用局部空间核的CNNs难以捕获长程依赖，除非使用大尺寸核，但这会带来显著的计算成本。为克服此限制，我们提出谱卷积条件神经过程（SConvCNPs），其在频域执行全局卷积。受用于学习偏微分方程（PDEs）解算子的傅里叶神经算子（FNOs）启发，我们的方法直接在频域参数化卷积核，利用了许多自然信号在傅里叶表示中相对紧凑且具有全局性的特性。我们在合成数据集和真实数据集上验证了SConvCNPs的有效性，展示了算子学习的思想如何能够推动NPs能力的进步。