mlsanityData to 14 Oct 2026

Uncertainty propagation in auto-regressive random neural network models

arXiv 2608.20483 ↗First posted 20 Aug 2026stat.ML cs.LG cs.NEcited by 0

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Abstract (arXiv)

We develop analytical and particle-based methods for uncertainty propagation in random neural network models, where both the inputs and network parameters are allowed to be random. Building on the piecewise-linear structure of the Leaky ReLU activation function, we derive a local approximation of the neural network output with respect to perturbations in both its inputs and parameters. This approximation is exact for perturbations that preserve the network activation pattern, and it allows us to compute analytical expressions for the probability density function and characteristic function of the network output, together with closed-form approximations for its mean and covariance. We extend this uncertainty propagation framework to autonomous dynamical systems whose one-step evolution map is represented by a random neural network. Repeated application of this map defines an autoregressive model, for which we derive recursive equations to propagate uncertainty in both the state and network parameters over time. These equations explicitly account for the state-parameter cross-covariance that develops under successive iterations of the network. Numerical experiments on the Lorenz-63 system and the Kuramoto-Sivashinsky equation demonstrate accurate uncertainty propagation through the predictability horizon and the applicability of the proposed framework to high-dimensional dynamical systems.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 28 of its 47 references that are in the map. First posted on arXiv 20 Aug 2026.

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References · 28

  1. 2015Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learningcited by 4,448
  2. 2016Simple and Scalable Predictive Uncertainty Estimation using Deep Ensemblescited by 3,926
  3. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,621
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  11. 2015Explaining NonLinear Classification Decisions with Deep Taylor Decompositioncited by 250
  12. 2017Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equationscited by 208
16 more
  1. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 170
  2. 2020Bayesian Neural Networks: An Introduction and Surveycited by 131
  3. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 118
  4. 2023Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operatorscited by 28
  5. 2023A Primer on Bayesian Neural Networks: Review and Debatescited by 27
  6. 2024Uncertainty Quantification via Stable Distribution Propagationcited by 10
  7. 2016Neural Taylor Approximations: Convergence and Exploration in Rectifier Networkscited by 9
  8. 2018Uncertainty Propagation in Deep Neural Networks Using Extended Kalman Filteringcited by 8
  9. 2024Physics-informed Neural Networks for Functional Differential Equations: Cylindrical Approximation and Its Convergence Guaranteescited by 5
  10. 2025HopCast: Calibration of Autoregressive Dynamics Modelscited by 3
  11. 2025Uncertainty propagation in feed-forward neural network modelscited by 3
  12. 2019Polynomial Neural Networks and Taylor maps for Dynamical Systems Simulation and Learningcited by 2
  13. 2023HOPE: High-order Polynomial Expansion of Black-box Neural Networkscited by 2
  14. 2025Surrogate modeling for uncertainty quantification in nonlinear dynamicscited by 1
  15. 2025Assumed Density Filtering and Smoothing with Neural Network Surrogate Modelscited by 1
  16. 2026Exact Gaussian Moment Matching for Residual Networks: a Second-Order Methodcited by 1

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