HDNet: Physics-Inspired Neural Network for Flow Estimation based on Helmholtz Decomposition

TL;DR

HDNet proposes a differentiable Helmholtz decomposition network that splits an arbitrary flow $\boldsymbol{v}^*$ into curl-free $\boldsymbol{v}_{irr}$ and divergence-free $\boldsymbol{v}_{sol}$ components via a learned scalar potential $\phi$, enabling hard physical constraints in flow reconstruction. It introduces Helmholtz synthesis to generate large, labeled training data from Perlin-noise–based scalar and vector fields, bypassing costly fluid simulations. Integrated into a PINN-based flow-reconstruction pipeline, HDNet enforces physical priors while maintaining differentiability, improving reconstruction accuracy and preserving physical properties in synthetic PIV and BOS experiments. The work demonstrates 2D feasibility and outlines clear paths to 3D, with potential wide applicability to inverse imaging, differential reconstruction, and forward simulations that require physically consistent vector fields.

Abstract

Flow estimation problems are ubiquitous in scientific imaging. Often, the underlying flows are subject to physical constraints that can be exploited in the flow estimation; for example, incompressible (divergence-free) flows are expected for many fluid experiments, while irrotational (curl-free) flows arise in the analysis of optical distortions and wavefront sensing. In this work, we propose a Physics- Inspired Neural Network (PINN) named HDNet, which performs a Helmholtz decomposition of an arbitrary flow field, i.e., it decomposes the input flow into a divergence-only and a curl-only component. HDNet can be trained exclusively on synthetic data generated by reverse Helmholtz decomposition, which we call Helmholtz synthesis. As a PINN, HDNet is fully differentiable and can easily be integrated into arbitrary flow estimation problems.

Figures (17)

• Figure 1: Overview of our pipeline. (a) Concept of HDNet: an arbitrary flow field $\mathbf{ v^*}$ is decomposed into its irrotational (curl free) and solenoidal (divergence free) components. This is known as a Helmholtz Decomposition (see text for details). (b) An example of how HDNet can be used to enforce the physical properties of vector fields in flow reconstruction problems. A motion MLP is trained to represent the flow of particles in a fluid. The output of the MLP is not necessarily physically plausible. Physical plausibility can be restored by utilizing HDNet to extract the incompressible (i.e. divergence free) component of the flow, which is then used to establish the temporal relationship between the particle fields over time (see Section \ref{['PIV2Sec:Experiments']}). (c) HDNet is trained exclusively on synthetic data, generated by a reverse process we term "Helmholtz Synthesis": irrotational and solenoidal components are synthesized from spatially smooth random scalar fields generated with Perlin Noise bridson2007curl. They can be combined into general flows $\mathbf{ v^*}$, forming training pairs with a known Helmholtz decomposition, without requiring costly fluid simulation.
• Figure 2: Perlin noise. The red grid in the Perlin noise subfigures controls the Perlin noise frequency. $\phi_1$ means with grid $2 \times 2$. More grid number means high frequency Perlin noise. (a) is Perlin noise for generating irrotational field. $\mathbf{\nabla}$ mean the gradient of Perlin noise. (b) is for solenoidal field $\mathbf{\nabla \times}$ mean the curl of Perlin noise.
• Figure 3: Synthetic PIV data reconstruction comparison. For each method we illustrate the flow field (Flow) and its divergence (DIV). (a) Ground Truth, (b-c) Horn-Schunck optical flow reconstruction, (d-e) our pipeline without HDNet, (f-g) our pipeline with HDNet. We also include quantitative evaluations: AAE (average angular error), and MSE (mean squared error of the divergence).
• Figure 4: Real PIV data reconstruction comparison. Soft constraint is the method that add a divergence penalty term to the total loss.
• Figure 5: BOS reconstruction comparison. Curl mean the curl value map of left figure. w HDNet scalar $\phi$ means our flow reconstruction pipeline with HDNet and the scalar output of HDNet, that clearly shows the hot air plume above the candle.