Physics-Informed Neural Networks for Brain Molecular Transport from MRI

The method described in the paper “Investigating molecular transport in the human brain from MRI with physics‑informed neural networks” by Bastian Zapf, Johannes Haubner, Miroslav Kuchta, Geir Ringstad, Per Kristian Eide, Kent‑Andre Mardal centres on the application of Physics-Informed Neural Networks (PINNs) to solve an inverse biomedical problem: estimating the diffusion coefficient of tracer molecules in the human brain using Magnetic Resonance Imaging (MRI) data,.

The core ideas of this methodology can be broken down into the following key components:

1. Mesh-Free Integration of Data and Physics

The fundamental concept is to combine deep learning with partial differential equations (PDEs) without the need for complex geometric meshing,.

Neural Network as a Function Approximator: Instead of using a computational mesh (as in the Finite Element Method, FEM), the method employs a fully connected neural network. This network takes spatial coordinates ( $x$ ) and time ( $t$ ) as inputs and outputs the tracer concentration $c(x, t)$ ,.
Inverse Problem Solving: The method simultaneously solves the forward problem (simulating diffusion) and the inverse problem (finding the unknown diffusion parameter). The unknown diffusion coefficient, $D$ , is treated as a trainable parameter alongside the network’s weights and biases.

2. The Compound Loss Function

The training of the neural network is driven by a composite loss function that balances two competing objectives:

Data Loss ( $J$ ): This measures the discrepancy between the neural network’s predictions and the actual noisy concentration measurements taken from MRI scans at specific time points (0, 7, 24, and 46 hours),.
PDE Loss ( $L_r$ ): This enforces the physical laws governing the system—specifically the diffusion equation ( $\frac{\partial c}{\partial t} = D\Delta c$ ). The method penalises the network whenever its output violates this physical law at a set of collocation points within the brain domain.

3. Strategies for Handling Noise and Instability

A significant portion of the methodology focuses on overcoming the challenges posed by noisy, temporally sparse MRI data, which often causes standard PINN formulations to fail (e.g., converging to a diffusion coefficient of zero),. The authors implement three critical adjustments:

Robust Norms ( $\ell_1$ vs $\ell_2$ ): Standard PINNs often use the Mean Squared Error ( $\ell_2$ -norm) for the PDE loss. However, this method finds that using the $\ell_1$ -norm makes the cost function less sensitive to outliers in the noisy MRI data, preventing the network from overfitting the noise,.
High PDE Weighting: To ensure accurate parameter recovery, the “residual” (the error in obeying the physics equation) must be kept small. The authors observe that assigning a significantly higher weight ( $w_r$ ) to the PDE loss term is necessary to force the network to respect the diffusion equation rather than just fitting the noisy data points,.
Parameterization of $D$ : To prevent the optimisation algorithm from getting stuck in local minima where the diffusion coefficient approaches zero, $D$ is parameterized using a sigmoid function. This constrains the coefficient within a physically realistic range (e.g., between 0.1 and 1.2 mm $^2$ h $^{-1}$ ) and helps avoid numerical instabilities,.

4. Adaptive Refinement

The method employs Residual-based Adaptive Refinement (RAR) to ensure the physics are obeyed throughout the entire space-time domain, not just where data exists.

Because the MRI data is sparse (available only at a few specific times), the neural network may satisfy the physics well at those times but violate it in the intervals between them.
The RAR technique iteratively adds new training points (collocation points) in regions where the PDE residual is high. This forces the network to learn the correct physical behaviour in the “gaps” between MRI scans, resulting in a solution that is consistent with the finite element approach,.

Analogy

To understand this method, imagine you are trying to draw the path of a ball thrown through the air, but you only have three blurry photos of the ball taken seconds apart.

Standard Deep Learning: You might draw a squiggly line that passes exactly through the centre of every blurry spot in the photos. This fits the “data” perfectly but looks physically impossible (the ball zig-zags).
Physics-Informed Neural Network (The Method): You are forced to draw the line using a “stiff” flexible ruler that naturally bends in a smooth parabolic arc (representing the laws of physics).
- The Loss Function: You try to bend the ruler so it passes close to the photos (Data Loss) without snapping the ruler (PDE Loss).
- High PDE Weighting: You decide that keeping the ruler un-snapped is more important than hitting the exact centre of every blurry photo.
- Adaptive Refinement: You notice the ruler looks weird in the gaps between photos, so you look closer at those specific empty spaces to ensure the curve remains smooth and realistic throughout the entire flight.

Physics-Informed Neural Networks for Brain Molecular Transport from MRI