# Probabilistic Numeric Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2010.10876 · HTML: https://mlsanity.com/paper/2010.10876/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Oct 2020.

## Built on by (most cited first, 10 of 10)

- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Space](https://mlsanity.com/paper/2310.02970/) (2023)
- [Steerable Partial Differential Operators for Equivariant Neural Networks](https://mlsanity.com/paper/2106.10163/) (2021)
- [The Hintons in your Neural Network: a Quantum Field Theory View of Deep Learning](https://mlsanity.com/paper/2103.04913/) (2021)
- [PolyNet: Polynomial Neural Network for 3D Shape Recognition with PolyShape Representation](https://mlsanity.com/paper/2110.07882/) (2021)
- [End-to-End Supervised Multilabel Contrastive Learning](https://mlsanity.com/paper/2307.03967/) (2023)
- [Algorithmic Differentiation for Automated Modeling of Machine Learned Force Fields](https://mlsanity.com/paper/2208.12104/) (2022)
- [AdS-GNN -- a Conformally Equivariant Graph Neural Network](https://mlsanity.com/paper/2505.12880/) (2025)
- [Continuous Spiking Graph Neural Networks](https://mlsanity.com/paper/2404.01897/) (2024)

## Abstract (arXiv)

Continuous input signals like images and time series that are irregularly sampled or have missing values are challenging for existing deep learning methods. Coherently defined feature representations must depend on the values in unobserved regions of the input. Drawing from the work in probabilistic numerics, we propose Probabilistic Numeric Convolutional Neural Networks which represent features as Gaussian processes (GPs), providing a probabilistic description of discretization error. We then define a convolutional layer as the evolution of a PDE defined on this GP, followed by a nonlinearity. This approach also naturally admits steerable equivariant convolutions under e.g. the rotation group. In experiments we show that our approach yields a 3× reduction of error from the previous state of the art on the SuperPixel-MNIST dataset and competitive performance on the medical time series dataset PhysioNet2012.