Beyond Points: Chebyshev Surrogates for Region Queries on Neural Fields

Neural fields provide compact and expressive representations of signals and geometry, mapping coordinates in a continuous input domain to output values. However, many applications require or benefit from reasoning over continuous regions of that domain rather than evaluating the network at individual coordinates. Existing approaches either compute a linear surrogate over each region and reduce it to a pair of output bounds, as in range analysis, or require dedicated architectures and training to answer region queries, such as integrals. Instead, we approximate a trained neural field over a region with a closed-form surrogate, obtained by propagating a Chebyshev polynomial through the network in a single forward pass. The output of each activation is refitted with a discrete cosine transform, which prevents the degree of the surrogate from growing across layers and exposes it as a knob, trading computation against approximation accuracy. Since it describes how the field varies within the region, it enables a broad range of queries directly on the field, without further network evaluations. This lets us locate the sub-regions where the field takes a given value, instead of only accepting, rejecting, or subdividing the region. Moreover, the coefficients of the surrogate give the integral of the field over the region in closed form. We apply our framework to mesh extraction on neural signed distance functions, and to anti-aliased, adaptive-resolution reconstruction of neural images, improving both speed and accuracy over sampling-based and range-analysis methods.