# VNODE: A Piecewise Continuous Volterra Neural Network

arXiv: https://arxiv.org/abs/2509.24659 · HTML: https://mlsanity.com/paper/2509.24659/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 31 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Builds on (most cited first, 20 of 20)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Inception-v4, Inception-ResNet and the Impact of Residual Connections on Learning](https://mlsanity.com/paper/1602.07261/) (2016)
- [MobileViT: Light-weight, General-purpose, and Mobile-friendly Vision Transformer](https://mlsanity.com/paper/2110.02178/) (2021)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [TinyViT: Fast Pretraining Distillation for Small Vision Transformers](https://mlsanity.com/paper/2207.10666/) (2022)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Benchmarking Neural Network Robustness to Common Corruptions and Surface Variations](https://mlsanity.com/paper/1807.01697/) (2018)
- [Global Second-order Pooling Convolutional Networks](https://mlsanity.com/paper/1811.12006/) (2018)
- [MALI: A memory efficient and reverse accurate integrator for Neural ODEs](https://mlsanity.com/paper/2102.04668/) (2021)
- [Neural Ordinary Differential Equation based Recurrent Neural Network Model](https://mlsanity.com/paper/2005.09807/) (2020)
- [Volterra Neural Networks (VNNs)](https://mlsanity.com/paper/1910.09616/) (2019)
- [GalaxyEdit: Large-Scale Image Editing Dataset with Enhanced Diffusion Adapter](https://mlsanity.com/paper/2411.13794/) (2024)
- [WaveNet-Volterra Neural Networks for Active Noise Control: A Fully Causal Approach](https://mlsanity.com/paper/2504.04450/) (2025)
- [E-ConvNeXt: A Lightweight and Efficient ConvNeXt Variant with Cross-Stage Partial Connections](https://mlsanity.com/paper/2508.20955/) (2025)

## Abstract (arXiv)

This paper introduces Volterra Neural Ordinary Differential Equations (VNODE), a piecewise continuous Volterra Neural Network that integrates nonlinear Volterra filtering with continuous time neural ordinary differential equations for image classification. Drawing inspiration from the visual cortex, where discrete event processing is interleaved with continuous integration, VNODE alternates between discrete Volterra feature extraction and ODE driven state evolution. This hybrid formulation captures complex patterns while requiring substantially fewer parameters than conventional deep architectures. VNODE consistently outperforms state of the art models with improved computational complexity as exemplified on benchmark datasets like CIFAR10 and Imagenet1K.