A Finite Element Method - Informed Neural Network For Uncertainty Quantification

Abstract

Sampling approaches for uncertainty quantification for real-world engineering problems are associated with large computational time and cost. This cost comes from the expensive deterministic simulation. Usage of surrogate models is a common way to overcome this issue in engineering applications. A conventional Neural Network (NN) can be used for building such surrogates. However, these neural networks are built based on input-output pairs. It is not possible to verify that the predicted output satisfies underlying physics. In this contribution, a physics-informed neural network based on a hybrid model of machine learning and classical Finite Element Method (FEM) is presented for forward propagation of uncertainty. The method uses FEM during both training and prediction stages. A surrogate model based on neural network for high dimensional problem is constructed by constraining the predictions of the neural network with the discretized partial differential equation of the system. During the training stage, the predicted solution from the FEM informed Neural Network(FEM-NN) is used to compute the residual using stiffness matrices and force vectors. This residual is used as a custom loss function from NN. This makes the whole training unsupervised as it does not require any output values. Hence, the need for expensive FEM solves is circumvented. The FEM-NN hybrid also gives an estimate of the accuracy of prediction by means of the calculated residual along with the prediction. The framework does not require mandatory expensive linear solves of the discretized equation instead substitutes the prediction from the neural network for computing the residual. This reduces the expensive training phase of the problem and can be applicable to real-world FEM simulations. The trained neural network is then sampled in a Monte Carlo (MC) manner to evaluate the statistics of the Quantities of Interest (QoI). The resulting FEM-NN hybrid is physics confirming and data-efficient. The efficacy of the framework is presented by a series of test case examples. The results are compared with classical MC results. The suitability of the method for the uncertainty quantification is studied and presented.