# Deep Decomposition for Stochastic Normal-Abnormal Transport

arXiv: https://arxiv.org/abs/2111.14777 · HTML: https://mlsanity.com/paper/2111.14777/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 72 references that are in the map.
- First posted on arXiv 29 Nov 2021.

## Builds on (most cited first, 20 of 20)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [FlowNet: Learning Optical Flow with Convolutional Networks](https://mlsanity.com/paper/1504.06852/) (2015)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Deep multi-scale video prediction beyond mean square error](https://mlsanity.com/paper/1511.05440/) (2015)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [Deep Fluids: A Generative Network for Parameterized Fluid Simulations](https://mlsanity.com/paper/1806.02071/) (2018)
- [Quicksilver: Fast Predictive Image Registration - a Deep Learning Approach](https://mlsanity.com/paper/1703.10908/) (2017)
- [Deep Learning for Physical Processes: Incorporating Prior Scientific Knowledge](https://mlsanity.com/paper/1711.07970/) (2017)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Trivializations for Gradient-Based Optimization on Manifolds](https://mlsanity.com/paper/1909.09501/) (2019)
- [SDE-Net: Equipping Deep Neural Networks with Uncertainty Estimates](https://mlsanity.com/paper/2008.10546/) (2020)
- [EikoNet: Solving the Eikonal equation with Deep Neural Networks](https://mlsanity.com/paper/2004.00361/) (2020)
- [Networks for Joint Affine and Non-parametric Image Registration](https://mlsanity.com/paper/1903.08811/) (2019)
- [Discovering Hidden Physics Behind Transport Dynamics](https://mlsanity.com/paper/2011.12222/) (2020)
- [Probabilistic Diffeomorphic Registration: Representing Uncertainty](https://mlsanity.com/paper/1701.03266/) (2017)

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

- [Brain-ID: Learning Contrast-agnostic Anatomical Representations for Brain Imaging](https://mlsanity.com/paper/2311.16914/) (2023)
- [PEPSI: Pathology-Enhanced Pulse-Sequence-Invariant Representations for Brain MRI](https://mlsanity.com/paper/2403.06227/) (2024)
- [A Modality-agnostic Multi-task Foundation Model for Human Brain Imaging](https://mlsanity.com/paper/2509.00549/) (2025)
- [Multi-delay arterial spin-labeled perfusion estimation with biophysics simulation and deep learning](https://mlsanity.com/paper/2311.10640/) (2023)

## Abstract (arXiv)

Advection-diffusion equations describe a large family of natural transport processes, e.g., fluid flow, heat transfer, and wind transport. They are also used for optical flow and perfusion imaging computations. We develop a machine learning model, D^2-SONATA, built upon a stochastic advection-diffusion equation, which predicts the velocity and diffusion fields that drive 2D/3D image time-series of transport. In particular, our proposed model incorporates a model of transport atypicality, which isolates abnormal differences between expected normal transport behavior and the observed transport. In a medical context such a normal-abnormal decomposition can be used, for example, to quantify pathologies. Specifically, our model identifies the advection and diffusion contributions from the transport time-series and simultaneously predicts an anomaly value field to provide a decomposition into normal and abnormal advection and diffusion behavior. To achieve improved estimation performance for the velocity and diffusion-tensor fields underlying the advection-diffusion process and for the estimation of the anomaly fields, we create a 2D/3D anomaly-encoded advection-diffusion simulator, which allows for supervised learning. We further apply our model on a brain perfusion dataset from ischemic stroke patients via transfer learning. Extensive comparisons demonstrate that our model successfully distinguishes stroke lesions (abnormal) from normal brain regions, while reconstructing the underlying velocity and diffusion tensor fields.