# PI-NOMT: Physics-Informed Neural Optimal Mass Transport for Brain Fluid Dynamics

arXiv: https://arxiv.org/abs/2609.33857 · HTML: https://mlsanity.com/paper/2609.33857/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 31 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 5 of 5)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Deep Variational Bayes Filters: Unsupervised Learning of State Space Models from Raw Data](https://mlsanity.com/paper/1605.06432/) (2016)
- [GlymphVIS: Visualizing Glymphatic Transport Pathways Using Regularized Optimal Transport](https://mlsanity.com/paper/1808.08304/) (2018)

## Abstract (arXiv)

Recovering hidden transport mechanisms from sparse spatiotemporal observations is a fundamental inverse problem in scientific machine learning. In brain tracer imaging, dynamic contrast-enhanced MRI (DCE-MRI) provides time-resolved measurements of tracer concentration, while the underlying velocity and source mechanisms governing tracer propagation remain unobserved. We formulate this problem as physics-informed latent-state inference, in which the transport field itself is the primary object of inference rather than an auxiliary variable used only to reconstruct observed densities. We propose Physics-Informed Neural Optimal Mass Transport (PI-NOMT), a framework that represents density, velocity, and source as continuous neural fields and combines a continuous neural density teacher, recursive differentiable advection--diffusion--source rollout, unbalanced optimal-transport regularization, and governing-equation supervision. Physical laws act as structural priors that constrain the space of admissible transport mechanisms, while observed tracer dynamics provide evidence for estimating the latent transport state. We evaluate PI-NOMT on a synthetic benchmark with known ground-truth transport and on DCE-MRI sequences from nine control rats. On the synthetic benchmark, PI-NOMT accurately recovers the prescribed velocity field, including its magnitude, direction, and integrated trajectories, rather than merely reconstructing endpoint densities. Across the nine rat datasets, the framework yields sub-percent local endpoint error, consistent physical speed scales, and low post-training PDE and incompressibility residuals. These results support physics-informed latent-state inference as a general framework for recovering hidden transport mechanisms from observed dynamic scalar fields.