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Uncertainty propagation in feed-forward neural network models.

Jeremy Diamzon¹, Daniele Venturi¹

¹Department of Applied Mathematics, UC Santa Cruz, Santa Cruz, CA, 95064, USA.

Neural Networks : the Official Journal of the International Neural Network Society

|October 11, 2025

Summary

This summary is machine-generated.

New methods accurately predict neural network output uncertainty. Linearizing leaky ReLU activation functions provides precise statistical results, even with significant input perturbations, validated by simulations.

Keywords:

MLP networks Trustworthiness of neural network models Uncertainty quantification

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Area of Science:

Artificial Intelligence
Machine Learning
Computational Mathematics

Background:

Feed-forward neural networks (FNNs) are susceptible to input uncertainties.
Quantifying output uncertainty in FNNs, especially with non-linear activation functions like leaky ReLU, remains challenging.
Existing methods often struggle with analytical tractability and accuracy under significant input perturbations.

Purpose of the Study:

To develop novel uncertainty propagation methods for FNNs with leaky ReLU activations.
To derive analytical expressions for the probability density function (PDF) and statistical moments of the FNN output.
To propose tractable surrogate models for approximating the joint PDF of the network output.

Main Methods:

Derivation of analytical expressions for output PDF and statistical moments considering input vector perturbations.
Application of a specific linearization technique for the leaky ReLU activation function.
Development of Gaussian copula surrogate models for approximating the output PDF.
Validation through Monte Carlo simulations and error analysis on a complex FNN model.

Main Results:

Accurate analytical expressions for output uncertainty and statistical moments were derived.
A key finding is the accuracy of a linearized leaky ReLU approach, even for large input perturbations.
Proposed Gaussian copula models provide analytically tractable approximations for the output PDF.
Theoretical predictions showed excellent agreement with Monte Carlo simulation results.

Conclusions:

The developed methods provide accurate and analytically tractable ways to propagate input uncertainty through FNNs with leaky ReLU activations.
The linearization technique is effective for handling significant input perturbations, simplifying uncertainty quantification.
The study offers valuable tools for understanding and managing uncertainty in complex neural network models.