A Mixed Residual Hybrid Method For Failure Probability Estimation

Solving partial differential equations (PDEs) with random input parameters via standard numerical schemes such as finite element methods is computationally expensive, especially when high-dimensional random parameters are involved. Evaluation of the failure probability involves massive repeated solving equations, which would be computationally prohibitive via traditional Monte Carlo methods. Using neural networks as a surrogate model can somewhat alleviate computational complexity. However, constructing a relatively accurate neural network requires a substantial number of labeled data for training. In this paper, we propose a new mixed residual hybrid (MRH) method for failure probability estimation. On the benefits of absorbing equation form into the loss function of neural networks, none of the labeled data is needed in the training phase. Expensive numerical methods shall not be used unless to correct the outputs in suspicious intervals. Compared to the traditional Monte Carlo method requiring millions of computations, numerical experiments demonstrated the efficiency of the MRH method, which only requires a few thousand calculations.