Proposal-Guided Greedy Surrogate Refinement for PDE-Driven High-Dimensional Rare-Event Estimation

2026-05-14

AI Summary

This work addresses the challenge of simulating rare events governed by high-dimensional partial differential equations, where constructing globally accurate surrogate models is hindered by the prohibitive cost of high-fidelity evaluations. The authors propose a surrogate-assisted adaptive importance sampling framework that employs a hybrid encoder–neural network architecture to generate low-dimensional latent representations. The surrogate model is updated locally along the evolving proposal distribution, and a greedy strategy in the latent space balances proximity to the failure boundary with sample diversity. Theoretical analysis provides error bounds on proposal stability, misclassification, and finite-sample estimation under local surrogate approximation errors. On multimodal and PDE-driven benchmark problems up to 100 dimensions, the method achieves accuracy comparable to that of adaptive importance sampling with the true model, using significantly fewer high-fidelity evaluations.

Abstract

Accurate surrogate construction for PDE-driven high-dimensional rare-event simulation is challenging when performance evaluations are expensive. Since a globally accurate surrogate may require many high-fidelity evaluations, adaptive importance sampling provides a natural localization tool: its evolving proposal distribution progressively identifies the failure-relevant region. Motivated by this observation, we propose a surrogate-assisted adaptive importance sampling framework that refines the surrogate locally along the evolving proposal, rather than over the entire input space. The surrogate combines an encoder with a neural network, providing a low-dimensional latent representation for both prediction and sample selection. At each adaptive iteration, candidates drawn from the current proposal are selected by a greedy latent-space rule balancing proximity to the estimated failure boundary and sample diversity. The selected samples are evaluated by the high-fidelity model and used to refine the surrogate, which then guides the subsequent cross-entropy-type adaptive proposal update. We establish one-step proposal stability bounds under local surrogate errors, together with surrogate-induced misclassification and finite-sample estimation error bounds. Numerical experiments on multimodal benchmarks and PDE-driven rare-event problems up to 100 dimensions show that the proposed method achieves accuracy comparable to true-model adaptive importance sampling while requiring substantially fewer high-fidelity evaluations.

Problem

• rare-event estimation
• surrogate modeling
• high-dimensional PDE
• adaptive importance sampling
• expensive evaluations

Innovation

• surrogate-assisted adaptive importance sampling
• proposal-guided refinement
• latent-space greedy selection
• PDE-driven rare-event estimation
• high-dimensional uncertainty quantification