# Variational volume reconstruction with the Deep Ritz Method

arXiv: https://arxiv.org/abs/2508.08309 · HTML: https://mlsanity.com/paper/2508.08309/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 45 references that are in the map.
- First posted on arXiv 8 Aug 2025.

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

- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [A Deep Cascade of Convolutional Neural Networks for MR Image Reconstruction](https://mlsanity.com/paper/1703.00555/) (2017)
- [A deep learning energy method for hyperelasticity and viscoelasticity](https://mlsanity.com/paper/2201.08690/) (2022)
- [Deep variational network for rapid 4D flow MRI reconstruction](https://mlsanity.com/paper/2004.09610/) (2020)
- [GBM Volumetry using the 3D Slicer Medical Image Computing Platform](https://mlsanity.com/paper/1303.0964/) (2013)
- [Fully Convolutional Slice-to-Volume Reconstruction for Single-Stack MRI](https://mlsanity.com/paper/2312.03102/) (2023)
- [Variational Image Segmentation Model Coupled with Image Restoration Achievements](https://mlsanity.com/paper/1405.2128/) (2014)
- [Super-Resolution Surface Reconstruction from Few Low-Resolution Slices](https://mlsanity.com/paper/2309.05071/) (2023)

## Built on by (most cited first, 2 of 2)

- [Finding geodesics with the Deep Ritz method](https://mlsanity.com/paper/2510.15177/) (2025)
- [Learning Gradient Flow: Using Equation Discovery to Accelerate Engineering Optimization](https://mlsanity.com/paper/2602.13513/) (2026)

## Abstract (arXiv)

We present a novel approach to variational volume reconstruction from sparse, noisy slice data using the Deep Ritz method. Motivated by biomedical imaging applications such as MRI-based slice-to-volume reconstruction (SVR), our approach addresses three key challenges: (i) the reliance on image segmentation to extract boundaries from noisy grayscale slice images, (ii) the need to reconstruct volumes from a limited number of slice planes, and (iii) the computational expense of traditional mesh-based methods. We formulate a variational objective that combines a regression loss designed to avoid image segmentation by operating on noisy slice data directly with a modified Cahn-Hilliard energy incorporating anisotropic diffusion to regularize the reconstructed geometry. We discretize the phase field with a neural network, approximate the objective at each optimization step with Monte Carlo integration, and use ADAM to find the minimum of the approximated variational objective. While the stochastic integration may not yield the true solution to the variational problem, we demonstrate that our method reliably produces high-quality reconstructed volumes in a matter of seconds, even when the slice data is sparse and noisy.