Physics-aligned Schr\"{o}dinger bridge

TL;DR

The paper introduces PalSB, a physics-aligned Schrödinger bridge framework that improves the accuracy and physical consistency of reconstructing complex physical fields from sparse data using a dual-stage training and boundary-aware sampling.

Contribution

It proposes a novel diffusion Schrödinger bridge model tailored to physical constraints, with a dual-stage training process and boundary-aware sampling for improved field reconstruction.

Findings

Achieves higher accuracy than existing methods.

Better compliance with physical boundary conditions.

Effective on complex nonlinear systems.

Abstract

The reconstruction of physical fields from sparse measurements is pivotal in both scientific research and engineering applications. Traditional methods are increasingly supplemented by deep learning models due to their efficacy in extracting features from data. However, except for the low accuracy on complex physical systems, these models often fail to comply with essential physical constraints, such as governing equations and boundary conditions. To overcome this limitation, we introduce a novel data-driven field reconstruction framework, termed the Physics-aligned Schr\"{o}dinger Bridge (PalSB). This framework leverages a diffusion Schr\"{o}dinger bridge mechanism that is specifically tailored to align with physical constraints. The PalSB approach incorporates a dual-stage training process designed to address both local reconstruction mapping and global physical principles. Additionally, a boundary-aware sampling technique is implemented to ensure adherence to physical boundary conditions. We demonstrate the effectiveness of PalSB through its application to three complex nonlinear systems: cylinder flow from Particle Image Velocimetry experiments, two-dimensional turbulence, and a reaction-diffusion system. The results reveal that PalSB not only achieves higher accuracy but also exhibits enhanced compliance with physical constraints compared to existing methods. This highlights PalSB's capability to generate high-quality representations of intricate physical interactions, showcasing its potential for advancing field reconstruction techniques.