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

arXiv: https://arxiv.org/abs/2406.08570 · HTML: https://mlsanity.com/paper/2406.08570/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 47 references that are in the map.
- First posted on arXiv 12 Jun 2024.

## Builds on (most cited first, 9 of 9)

- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [WIRE: Wavelet Implicit Neural Representations](https://mlsanity.com/paper/2301.05187/) (2023)
- [Physion: Evaluating Physical Prediction from Vision in Humans and Machines](https://mlsanity.com/paper/2106.08261/) (2021)
- [Accelerating Eulerian Fluid Simulation With Convolutional Networks](https://mlsanity.com/paper/1607.03597/) (2016)
- [Phase Retrieval via Wirtinger Flow: Theory and Algorithms](https://mlsanity.com/paper/1407.1065/) (2014)
- [Adaptive Neural Network-Based Approximation to Accelerate Eulerian Fluid Simulation](https://mlsanity.com/paper/2008.11832/) (2020)
- [Dynamic Structured Illumination Microscopy with a Neural Space-time Model](https://mlsanity.com/paper/2206.01397/) (2022)

## Built on by (most cited first, 4 of 4)

- [Closing the Approximation Gap in Simulation-free Latent SDEs](https://mlsanity.com/paper/2606.16138/) (2026)
- [ReynoldsFlow: Physics-Inspired Spatiotemporal Flow Representation for Video Understanding](https://mlsanity.com/paper/2503.04500/) (2025)
- [Physically Typed and Geometry-Aware Representations for Earth Foundation Models](https://mlsanity.com/paper/2609.13868/) (2026)
- [Learning Incompressible 3D Flows from Sparse Samples: A Set Transformer Approach to Divergence-Free Kernel Fitting](https://mlsanity.com/paper/2610.16645/) (2026)

## Abstract (arXiv)

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.