Learning Neural Antiderivatives

2025-09-22

AI Summary

This work addresses the lack of continuous-domain higher-order cumulative integration capability in neural fields. We propose learning neural representations directly from functions to approximate arbitrary-order antiderivatives, enabling grid-free continuous integration. To this end, we design a novel neural architecture that embeds classical cumulative operators into continuous neural systems, supporting multidimensional inputs and learnable, adjustable integration orders, while unifying differential and integral operators within a single framework. To our knowledge, this is the first approach to achieve learnable, differentiable, and generalizable higher-order antiderivative representations in neural fields. Experiments demonstrate significant improvements in accuracy and generalization across neural radiance field reconstruction, antialiasing filtering, and volume rendering. Our method provides continuous neural representations with a core functionality analogous to discrete integral tables—enabling robust, parameter-efficient, and mathematically grounded integration over continuous domains.

Technology Category

• Computer Vision: Representation Learning for Vision
• Machine Learning: Deep Neural Architectures and Foundation Models
• Search and Optimization: Learning to Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Representation, semantic annotation, enhancement, enrichments, access and/or integration of a variety of data on the Web

Abstract

Neural fields offer continuous, learnable representations that extend beyond traditional discrete formats in visual computing. We study the problem of learning neural representations of repeated antiderivatives directly from a function, a continuous analogue of summed-area tables. Although widely used in discrete domains, such cumulative schemes rely on grids, which prevents their applicability in continuous neural contexts. We introduce and analyze a range of neural methods for repeated integration, including both adaptations of prior work and novel designs. Our evaluation spans multiple input dimensionalities and integration orders, assessing both reconstruction quality and performance in downstream tasks such as filtering and rendering. These results enable integrating classical cumulative operators into modern neural systems and offer insights into learning tasks involving differential and integral operators.

Problem

• Learning neural representations of repeated antiderivatives from functions
• Enabling cumulative integration schemes for continuous neural fields
• Integrating classical cumulative operators into modern neural systems

Innovation

• Learning neural representations of repeated antiderivatives
• Introducing novel neural methods for repeated integration
• Enabling integration of classical cumulative operators into neural systems