# Neural Integro-Differential Equations

arXiv: https://arxiv.org/abs/2206.14282 · HTML: https://mlsanity.com/paper/2206.14282/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 52 references that are in the map.
- First posted on arXiv 28 Jun 2022.

## Builds on (most cited first, 7 of 7)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [An unsupervised deep learning approach in solving partial integro-differential equations](https://mlsanity.com/paper/2006.15012/) (2020)
- [Go with the FLOW: Visualizing spatiotemporal dynamics in optical widefield calcium imaging](https://mlsanity.com/paper/2009.14283/) (2020)

## Built on by (most cited first, 17 of 17)

- [Neural Integral Equations](https://mlsanity.com/paper/2209.15190/) (2022)
- [From Fourier to Neural ODEs: Flow Matching for Modeling Complex Systems](https://mlsanity.com/paper/2405.11542/) (2024)
- [Feedback Favors the Generalization of Neural ODEs](https://mlsanity.com/paper/2410.10253/) (2024)
- [Learning PDE Solution Operator for Continuous Modeling of Time-Series](https://mlsanity.com/paper/2302.00854/) (2023)
- [CaLMFlow: Volterra Flow Matching using Causal Language Models](https://mlsanity.com/paper/2410.05292/) (2024)
- [Spectral methods for Neural Integral Equations](https://mlsanity.com/paper/2312.05654/) (2023)
- [Projection Methods for Operator Learning and Universal Approximation](https://mlsanity.com/paper/2406.12264/) (2024)
- [Universal Approximation of Operators with Transformers and Neural Integral Operators](https://mlsanity.com/paper/2409.00841/) (2024)
- [Learning low-dimensional dynamics from whole-brain data improves task capture](https://mlsanity.com/paper/2305.14369/) (2023)
- [Operator Learning Meets Numerical Analysis: Improving Neural Networks through Iterative Methods](https://mlsanity.com/paper/2310.01618/) (2023)
- [Leray-Schauder Mappings for Operator Learning](https://mlsanity.com/paper/2410.01746/) (2024)
- [Neural delay differential equations: learning non-Markovian closures for partially known dynamical systems](https://mlsanity.com/paper/2410.02843/) (2024)
- [A Unified Representation of Neural Networks Architectures](https://mlsanity.com/paper/2512.17593/) (2025)
- [Neural Discovery of Memory and Nonlocal Kernels in Integro-Differential Equations with Constrained Kolmogorov--Arnold Networks](https://mlsanity.com/paper/2607.11110/) (2026)
- [DNA-SE: Towards Deep Neural-Nets Assisted Semiparametric Estimation](https://mlsanity.com/paper/2408.02045/) (2024)
- [Neural Integral Operators for Inverse Problems: An Operator-Learning Framework for Small-Sample Spectroscopic Classification](https://mlsanity.com/paper/2505.03677/) (2025)
- [Nonlocal operator learning for fMRI encoding and decoding tasks](https://mlsanity.com/paper/2605.20389/) (2026)

## Abstract (arXiv)

Modeling continuous dynamical systems from discretely sampled observations is a fundamental problem in data science. Often, such dynamics are the result of non-local processes that present an integral over time. As such, these systems are modeled with Integro-Differential Equations (IDEs); generalizations of differential equations that comprise both an integral and a differential component. For example, brain dynamics are not accurately modeled by differential equations since their behavior is non-Markovian, i.e. dynamics are in part dictated by history. Here, we introduce the Neural IDE (NIDE), a novel deep learning framework based on the theory of IDEs where integral operators are learned using neural networks. We test NIDE on several toy and brain activity datasets and demonstrate that NIDE outperforms other models. These tasks include time extrapolation as well as predicting dynamics from unseen initial conditions, which we test on whole-cortex activity recordings in freely behaving mice. Further, we show that NIDE can decompose dynamics into their Markovian and non-Markovian constituents via the learned integral operator, which we test on fMRI brain activity recordings of people on ketamine. Finally, the integrand of the integral operator provides a latent space that gives insight into the underlying dynamics, which we demonstrate on wide-field brain imaging recordings. Altogether, NIDE is a novel approach that enables modeling of complex non-local dynamics with neural networks.