Physics-aligned Schrödinger bridge

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 Schrödinger Bridge (PalSB). This framework leverages a diffusion Schrödinger 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.

Figures (17)

• Figure 1: Work flow of PalSB. In pretraining stage (top row), the low-fidelity field $\mathbf{y}$ is first interpolated to the same grid for output and the Gaussian-perturbed linear interpolation of the paired samples is then fed into the neural network to make a prediction of the high-fidelity output, where the residual between predition and label is utilized to optimize the neural network. In finetuning stage (bottom row), leveraging the pretrained model, a prediction of high-fidelity field is sampled from the low-fidelity condition through the DSB, assessed then by two metrics that evaluate the physical loss and regression loss. Subsequently, the model is tuned through the sampling path using the weighted loss.
• Figure 2: Visual comparison between different methods on the two varying tasks. In the first task (FI, top row), the low-fidelity observation (denoted as obs.) is 8x down-sampled from high-fidelity field of Kolmogorov flow on 256$\times$256 grid. In the second task (RI, bottom row), the observation is randomly sampled from high-fidelity field with 99% of the field masked. Our proposed method (SB and PalSB) visually outperforms other baselines that better recovers the spatial patterns as compared to the reference (denoted as ref.)
• Figure 3: Efficacy of boundary-aware sampling strategy
• Figure 4: Boundary-aware sampling strategy
• Figure 5: Early-stop sampling strategy