Session: 12-03-02: Data-Enabled Predictive Modeling, Scientific Machine Learning, and Uncertainty Quantification in Computational Mechanics

Paper Number: 113365

113365 - Physics Informed Neural Networks for Uncertainty Propagation for Alleviating the Curse of Dimensionality 

The reliability of engineering systems and components requires accurate and efficient computation of the probability of failure of response quantities that characterize structural performance. The increasing reliance of model-based design implies that it is necessary to quantify system uncertainties and propagate these uncertainties through the computational models predicting structural response. Uncertainty propagation in this setting is defined as the problem of predicting response statistics, including the probability distribution, of the solution variable of physics-based models given input uncertainty, which may arise from coefficients, source terms, or boundary and initial conditions. The traditional approach of uncertainty propagation includes characterizing input uncertainty, generating samples, and solving an ensemble of deterministic problems. The probability distribution of the response is then computed through statistical operations on the collected ensemble. This approach is fundamentally limited by the curse of dimensionality, which states that computational complexity increases exponentially with response to the stochastic dimension of the problem. Further, as computational models increase in size, including incorporating data containing uncertainties, the stochastic dimension of physics-based problems increases with the realism of the physical representation. Therefore, novel approaches are necessary that can perform uncertainty propagation with less computational complexity.

 

Physics informed neural networks (PINNs) are neural networks that take as inputs spatial-temporal coordinates as well as system parameters and output the solution field of a partial differential equation (PDE). Thus, they are surrogate models representing the solution field of physical response quantities, and they are determined by minimizing a loss function which describes its error in satisfying the definition of the PDEs. PINNS have gain tremendous interest in recent years because they have been demonstrated to accurately solve for the solution field for numerous PDEs. Since PINNs provide a direct mapping from input spatial-temporal coordinates to the solution field, they circumvent the need for traditional mesh-based discretization. Consequentially, they have the potential to overcome classical limitations of mesh-based representation of the solution field such as handling discontinuities or moving boundaries. However, in spite of neural networks being universal function approximators, the convergence of PINNs is often prohibitively slow. This has led to variations of the original PINN formulation including weak form PINNs (WF-PINNs), where the weak form of the PDE is utilized for the neural network loss function in lieu of the strong form. With WF-PINNs a mesh discretization can be constructed and the nodal coordinates are mapped to the solution field vector through the neural network. The choice of using the nodal coordinate is arbitrary and other parameters can be used otherwise. This work seeks to establish a mapping from stochastic collocation (SC) points to the solution vector within the PINN framework. Specifically, all the SC points necessary to build the surrogate model of the solution field over the sample space are assembled into the training dataset to optimize a neural network that maps the SC points to their corresponding solution vectors. This implies that the solution fields for all SC points are solved simultaneously. In contrast, de facto uncertainty propagation methods solve for each solution field as an independent, deterministic boundary value problem, leading to the so-called curse of dimensionality. The proposed approach, denoted by SC-PINNs, has the potential to circumvent the curse of dimensionality by solving for all SC points simultaneously. In this presentation, the formulation is described and demonstrated on numerical examples.

Presenting Author: Kirubel Teferra NRL

Presenting Author Biography: Dr. Kirubel Teferra is a mechanical engineer at the US Naval Research Laboratory (NRL) in the Materials Science & Technology Division since 2015. In 2019 he became a section head within the Multifunctional Materials Branch where he manages a group of 5 researchers. He works on a breadth of applications pertaining to Computational Mechanics with a focus on theory and algorithm development to improve predictability and reliability of simulations. Areas he has worked on include micromechanics, biomechanics, and probabilistic methods. He received his PhD from the department of Civil Engineering and Engineering Mechanics at Columbia University in 2011. Before joining NRL he held positions as a research engineering at Weidlinger Associates and a postdoctoral fellow in the department of Civil Engineering at Johns Hopkins University.

Authors:

Kirubel Teferra NRL