Augmented Neural Network Surrogate Models for Polynomial Chaos Expansions and Reduced Order Modeling

Mathematical models describing real world processes are becoming increasingly complex to better match the dynamics of the true system. While this is a positive step towards more complete knowledge of our world, numerical evaluations of these models become increasingly computationally inefficient, requiring increased resources or time to evaluate. This has led to the need for simplified surrogates to these complex mathematical models.

A growing surrogate modeling solution is with the usage of neural networks. Neural networks (NN) are known to generalize an approximation across a diverse dataset and minimize the solution along complex nonlinear boundaries. Additionally, these surrogate models can be found using only incomplete knowledge of the true dynamics. However, NN surrogates often suffer from a lack of interpretability, where the decisions made in the training process are not fully understood, and the roles of individual neurons are not well defined.

We present two solutions towards this lack of interpretability. The first focuses on mimicking polynomial chaos (PC) modeling techniques, modifying the structure of a NN to produce polynomial approximations of the underlying dynamics. This methodology allows for an extractable meaning from the network and results in improvement in accuracy over traditional PC methods. Secondly, we examine the construction of a reduced order modeling scheme using NN autoencoders, guiding the decisions of the training process to better match the real dynamics. This guiding process is performed via a physics-informed (PI) penalty, resulting in a speed-up in training convergence, but still results in poor performance compared to traditional schemes.