# Neural Integral Equations

**Authors:** Emanuele Zappala, Antonio Henrique de Oliveira Fonseca, Josue Ortega Caro, David van Dijk  
**arXiv:** [2209.15190](https://arxiv.org/abs/2209.15190)  
**Date:** 2022-09-30  
**Summary page:** https://www.summarizepaper.com/en/paper/2209.15190v1/

## Key points

- Integral equations (IEs) are functional equations where the unknown function is integrated over a potentially multidimensional space.
- IEs have significant applications in theoretical and applied sciences, including physics, chemistry, biology, and engineering.
- They are valuable for tackling inverse problems and modeling non-local dynamical systems with long-distance spatiotemporal relations.
- IEs offer an alternative formulation to differential equations like ODEs and PDEs that can be more convenient to solve.
- IEs can represent inherently non-local systems that traditional ODEs and PDEs may struggle to capture effectively.
- A novel approach called Neural Integral Equations (NIE) has been introduced to learn integral equations and their associated dynamics solely from data.
- An enhanced version known as Attentional Neural Integral Equations (ANIE) replaces the traditional integral operator with self-attention mechanisms for scalability and interpretability.
- Learning dynamics through integral equations using NIE is faster than other continuous methods such as Neural ODEs.
- ANIE demonstrates superior performance on benchmark tasks involving ODEs, PDEs, and IE systems using synthetic and real-world datasets.

## Summary

In the realm of mathematical equations, integral equations (IEs) stand out as functional equations where the unknown function is integrated over a potentially multidimensional space. These equations have found significant applications in various fields of theoretical and applied sciences, including physics, chemistry, biology, and engineering. They are particularly valuable for tackling inverse problems and modeling non-local dynamical systems with long-distance spatiotemporal relations. While differential equations like ordinary differential equations (ODEs) and partial differential equations (PDEs) are commonly used in scientific studies, IEs offer an alternative formulation that can be more convenient to solve. Moreover, IEs have the unique capability to represent inherently non-local systems that traditional ODEs and PDEs may struggle to capture effectively. Despite the existence of efficient algorithms for solving specific IEs, there has been a notable gap in methods capable of learning integral equations and their associated dynamics solely from data. To address this challenge, a novel approach called Neural Integral Equations (NIE) has been introduced. This method leverages a solver to learn unknown integral operators from data. Furthermore, an enhanced version known as Attentional Neural Integral Equations (ANIE) has been developed. In ANIE, the traditional integral operator is replaced by self-attention mechanisms, enhancing scalability and providing interpretability to the model. The study demonstrates that learning dynamics through integral equations using NIE is faster than other continuous methods such as Neural ODEs. The effectiveness of ANIE is showcased through superior performance on various benchmark tasks involving ODEs, PDEs, and IE systems using both synthetic and real-world data sets. With 10 + 12 pages comprising 12 figures and 7 tables,this work invites feedback from the scientific community to further advance understanding in this domain.