Reliability Analysis of Complex Systems using Subset Simulations with Hamiltonian Neural Networks

TL;DR

The paper tackles efficient reliability analysis for complex, high-dimensional systems by integrating Hamiltonian Neural Networks with Subset Simulation (HNNMC). By pretraining HNNs to predict Hamiltonian gradients, the method accelerates Hamiltonian Monte Carlo trajectories, delivering substantial speed-ups over traditional HMC, especially when model evaluations are costly or must be inferred from data. The authors demonstrate high accuracy across degenerate Gaussian, Rosenbrock, and Bouc-Wen Bayesian problems, and show that online error monitoring and occasional retraining mitigate limitations in tail regions and very high-dimensional spaces. The approach is particularly impactful for Bayesian inference workflows where gradient evaluations are expensive, enabling practical probabilistic reliability assessments. Overall, HNNMC offers a scalable, physics-informed surrogate strategy to enable rapid, robust rare-event estimation in complex engineering systems, with clear guidance on when to employ monitoring and retraining to preserve accuracy.

Abstract

We present a new Subset Simulation approach using Hamiltonian neural network-based Monte Carlo sampling for reliability analysis. The proposed strategy combines the superior sampling of the Hamiltonian Monte Carlo method with computationally efficient gradient evaluations using Hamiltonian neural networks. This combination is especially advantageous because the neural network architecture conserves the Hamiltonian, which defines the acceptance criteria of the Hamiltonian Monte Carlo sampler. Hence, this strategy achieves high acceptance rates at low computational cost. Our approach estimates small failure probabilities using Subset Simulations. However, in low-probability sample regions, the gradient evaluation is particularly challenging. The remarkable accuracy of the proposed strategy is demonstrated on different reliability problems, and its efficiency is compared to the traditional Hamiltonian Monte Carlo method. We note that this approach can reach its limitations for gradient estimations in low-probability regions of complex and high-dimensional distributions. Thus, we propose techniques to improve gradient prediction in these particular situations and enable accurate estimations of the probability of failure. The highlight of this study is the reliability analysis of a system whose parameter distributions must be inferred with Bayesian inference problems. In such a case, the Hamiltonian Monte Carlo method requires a full model evaluation for each gradient evaluation and, therefore, comes at a very high cost. However, using Hamiltonian neural networks in this framework replaces the expensive model evaluation, resulting in tremendous improvements in computational efficiency.

Figures (11)

• Figure 1: Comparison of the neural network architectures: (a) Scheme of a standard feedforward neural network; (b) Hamiltonian Neural Network architecture with an in-graph gradient, cf. Greydanus2019; (c) Scheme of latent Hamiltonian Neural Networks, cf. Dhulipala2022.
• Figure 2: Application of HNNMC for a bimodal Gaussian mixture distribution. (a) Trajectories in phase space of constant Hamiltonian. The trajectories are simulated using the gradients of the Hamiltonian (HMC) or the predicted values (HNN); (b) The distribution of $5000$ samples using the HNNMC compared with the true probability density function.
• Figure 3: Two-dimensional correlated Gaussian distribution: (a) $5000$ generated samples along with a long trajectory that moves through the distribution using traditional and HNN predicted gradients. (b) Probability density of the variables using $5000$ samples compared to the true probability density function.
• Figure 4: Illustration of the reliability problem for a bi-variate normal distribution with linear limit state function for different correlations: (a) $\rho=0$; (b) $\rho=0.75$; (c) $\rho=0.95$. (d) Illustration of the samples generated from Subset Simulation using the proposed HNNMC for a two-dimensional uncorrelated standard normal random vector having linear limit state function with $\beta=5$.
• Figure 5: Results of 100 Subset Simulations using MMH and HNNMC for correlated Gaussian distributions: the mean reliability indices are compared in the upper plots while the lower plots show the coefficient of variation for (a) $\beta = 3$, (b) $\beta = 4$, and (c) $\beta = 5$.