# Neural Controlled Differential Equations with Quantum Hidden Evolutions

arXiv: https://arxiv.org/abs/2404.19673 · HTML: https://mlsanity.com/paper/2404.19673/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 30 references that are in the map.
- First posted on arXiv 30 Apr 2024.

## Builds on (most cited first, 24 of 24)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Combining Recurrent, Convolutional, and Continuous-time Models with Linear State-Space Layers](https://mlsanity.com/paper/2110.13985/) (2021)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [GRU-ODE-Bayes: Continuous modeling of sporadically-observed time series](https://mlsanity.com/paper/1905.12374/) (2019)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Neural Rough Differential Equations for Long Time Series](https://mlsanity.com/paper/2009.08295/) (2020)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [Modeling Irregular Time Series with Continuous Recurrent Units](https://mlsanity.com/paper/2111.11344/) (2021)
- [Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise](https://mlsanity.com/paper/1906.02355/) (2019)
- [Trivializations for Gradient-Based Optimization on Manifolds](https://mlsanity.com/paper/1909.09501/) (2019)
- [Efficient and Accurate Gradients for Neural SDEs](https://mlsanity.com/paper/2105.13493/) (2021)
- [Neural ODE Processes](https://mlsanity.com/paper/2103.12413/) (2021)
- [UnICORNN: A recurrent model for learning very long time dependencies](https://mlsanity.com/paper/2103.05487/) (2021)
- [Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamics](https://mlsanity.com/paper/2110.10249/) (2021)
- [Neural Controlled Differential Equations for Online Prediction Tasks](https://mlsanity.com/paper/2106.11028/) (2021)
- [Heavy Ball Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.04840/) (2021)
- [The universal approximation theorem for complex-valued neural networks](https://mlsanity.com/paper/2012.03351/) (2020)
- [projUNN: efficient method for training deep networks with unitary matrices](https://mlsanity.com/paper/2203.05483/) (2022)
- [Stochasticity in Neural ODEs: An Empirical Study](https://mlsanity.com/paper/2002.09779/) (2020)
- [Theory and Implementation of Complex-Valued Neural Networks](https://mlsanity.com/paper/2302.08286/) (2023)
- [Complex Evolution Recurrent Neural Networks (ceRNNs)](https://mlsanity.com/paper/1906.02246/) (2019)

## Abstract (arXiv)

We introduce a class of neural controlled differential equation inspired by quantum mechanics. Neural quantum controlled differential equations (NQDEs) model the dynamics by analogue of the Schr"odinger equation. Specifically, the hidden state represents the wave function, and its collapse leads to an interpretation of the classification probability. We implement and compare the results of four variants of NQDEs on a toy spiral classification problem.