# Local Observability of a Class of Feedforward Neural Networks

## TL;DR

This paper explores the local observability of two-layer ReLU neural networks, identifying conditions for observability, proposing input design methods, and validating findings through numerical examples.

## Contribution

It provides the first analysis of local observability in FNNs, offering conditions for when weights are observable and methods to enhance observability.

## Key findings

- Weights are generally not locally observable.
- Certain network structures and weights enable local observability.
- Input design can improve weight distinguishability.

## Abstract

Beyond the traditional neural network training methods based on gradient descent and its variants, state estimation techniques have been proposed to determine a set of ideal weights from a control-theoretic perspective. Hence, the concept of observability becomes relevant in neural network training. In this paper, we investigate local observability of a class of two-layer feedforward neural networks~(FNNs) with rectified linear unit~(ReLU) activation functions. We analyze local observability of FNNs by evaluating an observability rank condition with respect to the weight matrix and the input sequence. First, we show that, in general, the weights of FNNs are not locally observable. Then, we provide sufficient conditions on the network structures and the weights that lead to local observability. Moreover, we propose an input design approach to render the weights distinguishable and show that this input also excites other weights inside a neighborhood. Finally, we validate our results through a numerical example.