# Fixed Integral Neural Networks

arXiv: https://arxiv.org/abs/2307.14439 · HTML: https://mlsanity.com/paper/2307.14439/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 8 references that are in the map.
- First posted on arXiv 26 Jul 2023.

## Builds on (most cited first, 4 of 4)

- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Reinforcement Learning with Deep Energy-Based Policies](https://mlsanity.com/paper/1702.08165/) (2017)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)

## Built on by (most cited first, 2 of 2)

- [Controlling Behavioral Diversity in Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2405.15054/) (2024)
- [2-Cats: 2D Copula Approximating Transforms](https://mlsanity.com/paper/2309.16391/) (2023)

## Abstract (arXiv)

It is often useful to perform integration over learned functions represented by neural networks. However, this integration is usually performed numerically, as analytical integration over learned functions (especially neural networks) is generally viewed as intractable. In this work, we present a method for representing the analytical integral of a learned function f. This allows the exact integral of a neural network to be computed, and enables constrained neural networks to be parametrised by applying constraints directly to the integral. Crucially, we also introduce a method to constrain f to be positive, a necessary condition for many applications (e.g. probability distributions, distance metrics, etc). Finally, we introduce several applications where our fixed-integral neural network (FINN) can be utilised.