Multiscale Structural Reliability Analysis in high dimensions with Tensor Trains and Physics-Augmented Neural Networks

2026-04-20

AI Summary

This work addresses the formidable computational challenges in high-dimensional, multiscale structural reliability analysis, where uncertainty propagation from microscale material randomness to macroscale responses renders conventional methods inefficient. The authors propose a novel framework that integrates a physics-informed Voigt–Reuss neural network (VRNN) with a deep inverse Rosenblatt transformation (DIRT). Under assumptions of scale separation and anisotropic linear elasticity, this approach uniquely combines the VRNN—enforcing physical consistency—with tensor-train-based high-dimensional importance sampling. The method enables efficient joint treatment of multiscale simulation and uncertainty quantification, achieving low-variance estimates of failure probabilities in benchmark three-dimensional heterogeneous material problems with up to 150 stochastic dimensions, thereby substantially enhancing computational efficiency and scalability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: Multimodal Perception & Sensor Fusion

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Structural reliability evaluation for composites constitutes a fundamentally high-dimensional multiscale problem, as microscale material uncertainties must propagate to the macroscale and can be quantified as high-dimensional random fields. Conventional approaches are computationally intractable, as they rely on repeatedly solving coupled partial differential equation systems across scales while contending with the exponential complexity inherent in high-dimensional uncertainty quantification. This work introduces a scalable and physically consistent framework that addresses both bottlenecks simultaneously in the case of separation of scales and (anisotropic) linear elasticity. In particular, we couple a physics-augmented Voigt--Reuss Neural Network (VRNN) with the Deep Inverse Rosenblatt Transport (DIRT) method to estimate the posterior probability of structural failure. The VRNN is used to resolve the computationally expensive FE$^2$ scheme by providing a near-instantaneous evaluation of the homogenized stiffness tensor that is guaranteed to be symmetric, positive-definite, and strictly bounded within the Voigt--Reuss limits, enabling fast evaluation of the homogenized responses. The DIRT method constructs a sequence of functional tensor train approximations to efficiently store an approximation of the high-dimensional optimal importance sampling distribution for estimating the probability of failure. This mitigates the curse of dimensionality arising from the Karhunen--Loève expansion of the random fields. The framework is demonstrated on a three-dimensional heterogeneous benchmark problem, where the uncertainty in the microscale material properties is characterized by a Bayesian posterior distribution obtained from limited strain observations. Our results show that the proposed framework can provide low-variance estimates of failure probabilities in dimensions up to 150.

Problem

• structural reliability
• high-dimensional uncertainty
• multiscale analysis
• random fields
• curse of dimensionality

Innovation

• Tensor Trains
• Physics-Augmented Neural Networks
• Multiscale Reliability Analysis
• High-Dimensional Uncertainty Quantification
• Deep Inverse Rosenblatt Transport