Principal Component Flow Map Learning of PDEs from Incomplete, Limited, and Noisy Data

TL;DR

This paper introduces a neural network-based method for modeling PDE dynamics from incomplete, noisy, and limited data, enabling efficient high-resolution simulations on high-dimensional, non-uniform grids.

Contribution

It presents a novel neural network architecture that models PDEs in a reduced basis, effective with limited and noisy data, improving efficiency and applicability in real-world scenarios.

Findings

Effective modeling of PDEs with limited, noisy data.

Significantly reduced neural network parameterization.

Enables rapid high-resolution PDE simulations.

Abstract

We present a computational technique for modeling the evolution of dynamical systems in a reduced basis, with a focus on the challenging problem of modeling partially-observed partial differential equations (PDEs) on high-dimensional non-uniform grids. We address limitations of previous work on data-driven flow map learning in the sense that we focus on noisy and limited data to move toward data collection scenarios in real-world applications. Leveraging recent work on modeling PDEs in modal and nodal spaces, we present a neural network structure that is suitable for PDE modeling with noisy and limited data available only on a subset of the state variables or computational domain. In particular, spatial grid-point measurements are reduced using a learned linear transformation, after which the dynamics are learned in this reduced basis before being transformed back out to the nodal space. This approach yields a drastically reduced parameterization of the neural network compared with previous flow map models for nodal space learning. This allows for rapid high-resolution simulations, enabled by smaller training data sets and reduced training times.