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WTNN: Weibull-Tailored Neural Networks for survival analysis

Gabrielle Rives, Olivier Lopez, Nicolas Bousquet

TL;DR

WTNN tackles survival analysis under censoring by learning covariate-dependent Weibull parameters η and β through a Weibull-tailored two-head neural network with a shared backbone. The architecture enforces isotonic consistency and Weibull-compatible constraints, supported by regularity penalties to improve identifiability and calibration. Through simulated and real-world military-vehicle data, WTNN delivers well-calibrated survival distributions and robust discrimination, outperforming competing deep survival models in calibration and stability. The work contributes a principled framework for integrating prior domain knowledge into neural models for system-level reliability with censored data, and outlines pathways for future Bayesian and competing-risks extensions.

Abstract

The Weibull distribution is a commonly adopted choice for modeling the survival of systems subject to maintenance over time. When only proxy indicators and censored observations are available, it becomes necessary to express the distribution's parameters as functions of time-dependent covariates. Deep neural networks provide the flexibility needed to learn complex relationships between these covariates and operational lifetime, thereby extending the capabilities of traditional regression-based models. Motivated by the analysis of a fleet of military vehicles operating in highly variable and demanding environments, as well as by the limitations observed in existing methodologies, this paper introduces WTNN, a new neural network-based modeling framework specifically designed for Weibull survival studies. The proposed architecture is specifically designed to incorporate qualitative prior knowledge regarding the most influential covariates, in a manner consistent with the shape and structure of the Weibull distribution. Through numerical experiments, we show that this approach can be reliably trained on proxy and right-censored data, and is capable of producing robust and interpretable survival predictions that can improve existing approaches.

WTNN: Weibull-Tailored Neural Networks for survival analysis

TL;DR

WTNN tackles survival analysis under censoring by learning covariate-dependent Weibull parameters η and β through a Weibull-tailored two-head neural network with a shared backbone. The architecture enforces isotonic consistency and Weibull-compatible constraints, supported by regularity penalties to improve identifiability and calibration. Through simulated and real-world military-vehicle data, WTNN delivers well-calibrated survival distributions and robust discrimination, outperforming competing deep survival models in calibration and stability. The work contributes a principled framework for integrating prior domain knowledge into neural models for system-level reliability with censored data, and outlines pathways for future Bayesian and competing-risks extensions.

Abstract

The Weibull distribution is a commonly adopted choice for modeling the survival of systems subject to maintenance over time. When only proxy indicators and censored observations are available, it becomes necessary to express the distribution's parameters as functions of time-dependent covariates. Deep neural networks provide the flexibility needed to learn complex relationships between these covariates and operational lifetime, thereby extending the capabilities of traditional regression-based models. Motivated by the analysis of a fleet of military vehicles operating in highly variable and demanding environments, as well as by the limitations observed in existing methodologies, this paper introduces WTNN, a new neural network-based modeling framework specifically designed for Weibull survival studies. The proposed architecture is specifically designed to incorporate qualitative prior knowledge regarding the most influential covariates, in a manner consistent with the shape and structure of the Weibull distribution. Through numerical experiments, we show that this approach can be reliably trained on proxy and right-censored data, and is capable of producing robust and interpretable survival predictions that can improve existing approaches.

Paper Structure

This paper contains 38 sections, 12 theorems, 100 equations, 6 figures, 15 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Notations and data
  3. Mission durations.
  4. Covariate description.
  5. Modeling and training
  6. Likelihood-based loss function
  7. Weibull-tailored neural networks
  8. Isotonic consistency with Weibull survival
  9. Base architecture description.
  10. From idealized theoretical properties to architectural choices
  11. Learning with additional regularity penalties
  12. Loss function for Assumption \ref{['assumption:risk_monotonicity']}
  13. Loss functions for Assumption \ref{['assump:distinct_heads']}
  14. Loss function to enforce statistical alignment
  15. Training and testing
  16. ...and 23 more sections

Key Result

Proposition 1

Denote $x\mapsto \Gamma(x)$ the gamma function defined on $\mathbb{R}^+_*$ and $\xi=\arg\min_{\mathbb{R}^+} \Gamma(x)$ ($\xi\simeq 3/2$). Denote $\beta_0=(\xi-1)^{-1}$. Under Assumption assumption:risk_monotonicity, then, $\forall \theta\in\Theta$,

Figures (6)

  • Figure 1: Typical distribution of vehicle availability and unavailability durations.
  • Figure 2: Neural network architecture.
  • Figure 3: Distribution in percentage of $L_1$ relative errors between simulated and real mission durations.
  • Figure 4: Predictive survival for the WTNN model (estimated for Fleet # 3).
  • Figure 5: Typical training evolution (training rate vs number of epochs).
  • ...and 1 more figures

Theorems & Definitions (24)

  • Proposition 1
  • Proposition 2: Uniform upper bound for $\eta$
  • Proposition 3: $C^2$ differentiability
  • Corollary 1: Existence and consistency of the MLE
  • Proposition 4: Regular Fisher information
  • Theorem 1: CLT for the MLE
  • Proposition 5: Architecture proposal
  • Proof : Proposition \ref{['prop:constraints']}
  • Proof : Proposition \ref{['prop:upper_bound_eta']}
  • Proof : Proposition \ref{['prop:C2-diff']}
  • ...and 14 more