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

2024-07-15

arXiv.org

AI Summary

Modeling partial differential equations (PDEs) on high-dimensional, non-uniform grids from partially observed, sparse, and noisy data remains challenging. Method: We propose a dynamics modeling framework that couples learnable principal component analysis (PCA)-based dimensionality reduction with flow graph neural networks. Crucially, we embed learnable linear dimensionality reduction directly into the flow graph learning pipeline, enabling dynamical modeling in a low-dimensional modal space while jointly learning the reconstruction mapping from modal coefficients to nodal solutions. The framework supports incomplete variable observations and irregular spatial sampling, and incorporates noise-robust training strategies. Contribution/Results: Our approach significantly reduces model parameter count, data requirements, and computational cost; achieves high predictive accuracy under sparse and noisy observation conditions; and enables high-resolution, fast PDE simulation. Experiments on photorealistic PDE simulation tasks demonstrate over 50% reduction in training time and more than 60% decrease in data demand compared to state-of-the-art methods.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Web Mining and Content Analysis: Large pretrained models with web data

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.

Problem

• Complex System Dynamics
• Partial Differential Equations
• Irregular Large Grids

Innovation

• Neural Network Methodology
• Sparse Data Learning
• Complex System Dynamics