# A Helmholtz equation solver using unsupervised learning: Application to transcranial ultrasound

arXiv: https://arxiv.org/abs/2010.15761 · HTML: https://mlsanity.com/paper/2010.15761/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Oct 2020.

## Built on by (most cited first, 12 of 12)

- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layers](https://mlsanity.com/paper/2306.17486/) (2023)
- [Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations](https://mlsanity.com/paper/2304.04234/) (2023)
- [Principled Acceleration of Iterative Numerical Methods Using Machine Learning](https://mlsanity.com/paper/2206.08594/) (2022)
- [A Learned Born Series for Highly-Scattering Media](https://mlsanity.com/paper/2212.04948/) (2022)
- [Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networks](https://mlsanity.com/paper/2503.20222/) (2025)
- [A Skull-Adaptive Framework for AI-Based 3D Transcranial Focused Ultrasound Simulation](https://mlsanity.com/paper/2505.12998/) (2025)
- [Synthetic Skull CT Generation with Generative Adversarial Networks to Train Deep Learning Models for Clinical Transcranial Ultrasound](https://mlsanity.com/paper/2308.00206/) (2023)
- [Physics-guided Full Waveform Inversion using Encoder-Solver Convolutional Neural Networks](https://mlsanity.com/paper/2405.17696/) (2024)
- [Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations](https://mlsanity.com/paper/2605.04853/) (2026)
- [McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation](https://mlsanity.com/paper/2606.30495/) (2026)
- [Neural Network Preconditioning: A Case Study for the Solution of the Parametric Helmholtz Equation](https://mlsanity.com/paper/2610.16542/) (2026)

## Abstract (arXiv)

Transcranial ultrasound therapy is increasingly used for the non-invasive treatment of brain disorders. However, conventional numerical wave solvers are currently too computationally expensive to be used online during treatments to predict the acoustic field passing through the skull (e.g., to account for subject-specific dose and targeting variations). As a step towards real-time predictions, in the current work, a fast iterative solver for the heterogeneous Helmholtz equation in 2D is developed using a fully-learned optimizer. The lightweight network architecture is based on a modified UNet that includes a learned hidden state. The network is trained using a physics-based loss function and a set of idealized sound speed distributions with fully unsupervised training (no knowledge of the true solution is required). The learned optimizer shows excellent performance on the test set, and is capable of generalization well outside the training examples, including to much larger computational domains, and more complex source and sound speed distributions, for example, those derived from x-ray computed tomography images of the skull.