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

Victor Churchill

TL;DR

The paper tackles learning the evolution of high-dimensional PDEs from partially observed, noisy data by introducing PC-FML, a reduced-basis flow-map framework. It jointly learns a linear reduced representation and a residual flow map in that reduced space, allowing large reductions in parameter count and training data needs while preserving predictive accuracy. The method provides fixed, constrained, or unconstrained options for the reduction and demonstrates superior stability and denoising performance across six challenging 1D and 2D PDEs, including Burgers, shallow water, wave, and Navier–Stokes equations. The approach enables rapid, locally executable high-resolution simulations on incomplete grids, with potential extensions to enforce conservation laws and physics-based constraints for further improvements.

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.

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

TL;DR

The paper tackles learning the evolution of high-dimensional PDEs from partially observed, noisy data by introducing PC-FML, a reduced-basis flow-map framework. It jointly learns a linear reduced representation and a residual flow map in that reduced space, allowing large reductions in parameter count and training data needs while preserving predictive accuracy. The method provides fixed, constrained, or unconstrained options for the reduction and demonstrates superior stability and denoising performance across six challenging 1D and 2D PDEs, including Burgers, shallow water, wave, and Navier–Stokes equations. The approach enables rapid, locally executable high-resolution simulations on incomplete grids, with potential extensions to enforce conservation laws and physics-based constraints for further improvements.

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.
Paper Structure (24 sections, 44 equations, 18 figures, 1 table)

This paper contains 24 sections, 44 equations, 18 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Literature Review
  4. Our focus and contributions
  5. Preliminaries
  6. FML of PDEs in Modal Space
  7. FML of Partially-Observed PDEs in Nodal Space
  8. Principal Component Flow Map Learning (PC-FML)
  9. Problem Setup and Data Collection
  10. Mathematical Motivation
  11. Reduced basis for nodal measurements
  12. Partial reduced dynamics
  13. Numerical Approach
  14. Determining reduced order dimension
  15. Model Training
  16. ...and 9 more sections

Figures (18)

  • Figure 2.1: FML model for nodal PDEs with missing information.
  • Figure 3.1: The proposed PC-FML network structure, with $N_{mem}=3$.
  • Figure 3.2: Determining reduced order dimension.
  • Figure 3.3: Ex. 5 (noiseless 2D wave): Reduced basis elements $b_j(\mathbf{x})$ for Fixed PC-FML.
  • Figure 3.4: Ex. 5 (noisy 2D wave): Reduced basis elements $b_j(\mathbf{x})$ for Fixed PC-FML.
  • ...and 13 more figures