# A Universal Approximation Result for Difference of log-sum-exp Neural Networks

## TL;DR

This paper proves that a specific class of neural networks, called Difference-LSE networks, can universally approximate continuous functions over convex sets and are particularly useful for optimization-based design tasks.

## Contribution

It introduces Difference-LSE networks as universal approximators with a difference-of-convex structure, enabling efficient optimization in design applications.

## Key findings

- Difference-LSE networks are universal approximators of continuous functions.
- They can be optimized efficiently using difference-of-convex algorithms.
- Application demonstrated in designing a diet for a type-2 diabetes patient.

## Abstract

We show that a neural network whose output is obtained as the difference of the outputs of two feedforward networks with exponential activation function in the hidden layer and logarithmic activation function in the output node (LSE networks) is a smooth universal approximator of continuous functions over convex, compact sets. By using a logarithmic transform, this class of networks maps to a family of subtraction-free ratios of generalized posynomials, which we also show to be universal approximators of positive functions over log-convex, compact subsets of the positive orthant. The main advantage of Difference-LSE networks with respect to classical feedforward neural networks is that, after a standard training phase, they provide surrogate models for design that possess a specific difference-of-convex-functions form, which makes them optimizable via relatively efficient numerical methods. In particular, by adapting an existing difference-of-convex algorithm to these models, we obtain an algorithm for performing effective optimization-based design. We illustrate the proposed approach by applying it to data-driven design of a diet for a patient with type-2 diabetes.