Revisiting Neural Processes via Fourier Transform and Volterra Series

Modeling unknown latent functions from finite, irregularly sampled measurements is a recurring challenge across science and engineering. Neural processes (NPs), a family of probabilistic functional models, are promising solutions---especially when endowed with domain-specific symmetries like translation equivariance, which improve sample efficiency and generalization. Yet existing translation-equivariant NPs face two limitations: (i) they stack generic components with non-linearities, obscuring the induced function class and limiting interpretability; and (ii) convolutional designs are limited by local receptive fields and the need to embed inputs onto a dense uniform grid, while attention-based alternatives lift these restrictions at quadratic cost in the number of observations. We address both with two contributions. First, using the Volterra expansion, we approximate continuous translation-equivariant operators by sums of higher-order convolutions, yielding analytical transparency while admitting efficient evaluation via first-order convolutions. Second, we introduce set Fourier convolutions (SFConvs), a frequency-domain parameterization that operates directly on irregularly sampled points, achieves approximately global receptive fields, and scales linearly in the number of observations. Building on these ideas, we propose two conditional NPs (CNPs): SFConvCNPs, which stack SFConv blocks with non-linearities, and SFVConvCNPs, which integrate the Volterra formulation. Experiments on synthetic and real-world datasets demonstrate our methods' efficacy against state-of-the-art baselines.

Scientists often need to reconstruct an unknown quantity—like temperature or fluid flow—from a handful of scattered, noisy measurements, while gauging how much to trust the gaps. Machine-learning tools called neural processes can do this. When the system has a known symmetry—such as patterns looking the same wherever you place your coordinate origin—building it into the model can improve accuracy and let it learn from fewer examples. But today's best symmetry-aware versions force a trade-off: one needs data on a dense, evenly spaced grid and only "sees" nearby points; another sees everything but grows prohibitively slow as measurements pile up—and both are opaque black boxes that resist mathematical analysis. We break this trade-off two ways. First, we use a classical mathematical tool (the Volterra series) to rewrite these models in an analytically transparent form, so design choices gain clear theoretical meaning—for example, how many blocks to stack reflects how much nonlinearity a task needs. Second, we design "set Fourier convolutions" that work directly on irregularly scattered points by computing in the frequency domain, weighing all measurements at once while cost grows only linearly with the data. Our two new models match or beat leading methods on tasks from image completion to simulating turbulence and predicting climate. The payoff: faster, more scalable predictions built on a more principled foundation for model design, even when scientific data are sparse and irregular.