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Energy Approach from $\varepsilon$-Graph to Continuum Diffusion Model with Connectivity Functional

Yahong Yang, Sun Lee, Jeff Calder, Wenrui Hao

TL;DR

This work develops an energy-based framework to rigorously derive a continuum diffusion model from an $\varepsilon$-graph endowed with a connectivity density $g$. It proves that the discrete graph energy converges to a nonlocal continuum energy with error $O(\varepsilon)$, where the bound depends only on the $W^{1,1}$-norm of $g$, making the result robust to sharp local fluctuations. The authors then connect this theory to brain dynamics by learning a spatially varying diffusivity $D(\mathbf{x})$ from edge-weight data via a neural network, and embedding the learned heterogeneity into a reaction-diffusion PDE on a brain domain. Numerical experiments on functional connectivity data demonstrate that the learned diffusion field yields dynamics that differ meaningfully from constant-diffusivity models, highlighting the importance of connectivity-informed heterogeneity. The framework combines a rigorous two-stage continuum limit (discrete to nonlocal to local) with a practical data-driven pipeline to recover diffusivity, offering a principled approach to connectivity-driven diffusion in neuroscience and related fields.

Abstract

We derive an energy-based continuum limit for $\varepsilon$-graphs endowed with a general connectivity functional. We prove that the discrete energy and its continuum counterpart differ by at most $O(\varepsilon)$; the prefactor involves only the $W^{1,1}$-norm of the connectivity density as $\varepsilon\to0$, so the error bound remains valid even when that density has strong local fluctuations. As an application, we introduce a neural-network procedure that reconstructs the connectivity density from edge-weight data and then embeds the resulting continuum model into a brain-dynamics framework. In this setting, the usual constant diffusion coefficient is replaced by the spatially varying coefficient produced by the learned density, yielding dynamics that differ significantly from those obtained with conventional constant-diffusion models.

Energy Approach from $\varepsilon$-Graph to Continuum Diffusion Model with Connectivity Functional

TL;DR

This work develops an energy-based framework to rigorously derive a continuum diffusion model from an -graph endowed with a connectivity density . It proves that the discrete graph energy converges to a nonlocal continuum energy with error , where the bound depends only on the -norm of , making the result robust to sharp local fluctuations. The authors then connect this theory to brain dynamics by learning a spatially varying diffusivity from edge-weight data via a neural network, and embedding the learned heterogeneity into a reaction-diffusion PDE on a brain domain. Numerical experiments on functional connectivity data demonstrate that the learned diffusion field yields dynamics that differ meaningfully from constant-diffusivity models, highlighting the importance of connectivity-informed heterogeneity. The framework combines a rigorous two-stage continuum limit (discrete to nonlocal to local) with a practical data-driven pipeline to recover diffusivity, offering a principled approach to connectivity-driven diffusion in neuroscience and related fields.

Abstract

We derive an energy-based continuum limit for -graphs endowed with a general connectivity functional. We prove that the discrete energy and its continuum counterpart differ by at most ; the prefactor involves only the -norm of the connectivity density as , so the error bound remains valid even when that density has strong local fluctuations. As an application, we introduce a neural-network procedure that reconstructs the connectivity density from edge-weight data and then embeds the resulting continuum model into a brain-dynamics framework. In this setting, the usual constant diffusion coefficient is replaced by the spatially varying coefficient produced by the learned density, yielding dynamics that differ significantly from those obtained with conventional constant-diffusion models.

Paper Structure

This paper contains 21 sections, 8 theorems, 110 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Weighted Graph Diffusion
  3. epsilon-Graph with Connectivity Density g
  4. Preliminaries
  5. Energy Approach
  6. Convergence from discrete energy to nonlocal energy
  7. Convergence from nonlocal energy to local energy
  8. Continuous Diffusion Equation
  9. Numerical Results
  10. Learning Spatially Varying Diffusivity via Neural Networks
  11. Neural Network Model for Learning Diffusivity
  12. Recovery of Handcrafted Diffusivity via Neural Network Training
  13. Training Procedure
  14. Domain Normalization
  15. Ground-Truth Function
  16. ...and 6 more sections

Key Result

Proposition 2.3

Suppose that $\Omega$ has a $C^{1,1}$ boundary, $g\in W^{1,\infty}(\Omega)$, $\rho\in L^1(\Omega)$, and Assumption assump:connectivity_functional holds. For any $\varepsilon>0$, $\bm{x}\in\Omega$ and $\lambda=\frac{\bar{g}}{\underline{g}}+1$ with $\varepsilon< \underline{g}\min\{r_\Omega/2\lambda, 1 where $B$ and $r_\Omega$ are constants only dependent on $\Omega$.

Figures (6)

  • Figure 1: Left: Axial slice from the 3D MRI volume used to construct the brain domain $\Omega$. Middle: Brain parcellation result. Right: Visualization of functional connectivity (FC) between brain parcels. Each parcel is represented by its centroid, and FC is shown as lines connecting parcel pairs. Line colors indicate connection strength.
  • Figure 2: Left: 3D brain domain reconstructed from MRI data. Middle: Computed diffusivity field $D(\bm{x})$ obtained from the trained neural network model $D(\bm{x}) = \frac{1}{g_\theta(\bm{x})^{d+2}}$, with $d = 3$. Right: The initial condition $u_0(\bm{x})$ was obtained from PET scan imaging.
  • Figure 3: Performance summary for the cosine-based ground-truth function with $\varepsilon = C \left({\log n}/{n}\right)^{1/(d+2)}$ chosen according to Eq. \ref{['varepsilon']} (Theorem \ref{['main1']}), for varying numbers of parcels $n$. Each experiment is repeated 100 times with independently sampled datasets. For each metric, we report the geometric mean across these repetitions, together with the multiplicative standard deviation, i.e. values of the form $\exp(\mu \pm \sigma)$ where $\mu$ and $\sigma$ denote the mean and standard deviation of the log-transformed results. Both axes are shown in logarithmic scale. For the definitions of Final Loss and Validation Loss, see Eq. \ref{['eq:loss_function_modified2']}; for the other metrics, refer to Eqs. \ref{['MAE']} and \ref{['RMAE']}.
  • Figure 4: Performance summary for the cosine-based ground-truth function with $\varepsilon = C \left({\log n}/{n}\right)^{1/(d+2)}$ chosen according to Eq. \ref{['varepsilon']} (Theorem \ref{['main1']}), for varying numbers of parcels $n$. Each experiment is repeated 100 times with independently sampled datasets. For each metric, we report the geometric mean across these repetitions, together with the multiplicative standard deviation, i.e. values of the form $\exp(\mu \pm \sigma)$ where $\mu$ and $\sigma$ denote the mean and standard deviation of the log-transformed results. Both axes are shown in logarithmic scale. For the definitions of Final Loss and Validation Loss, see Eq. \ref{['eq:loss_function_modified2']}; for the other metrics, refer to Eqs. \ref{['MAE']} and \ref{['RMAE']}.
  • Figure 5: Left: Time evolution of the difference in the boundary integral between the solutions computed using the trained diffusion coefficient $D(x)$ and a constant diffusion coefficient. Middle: $u_{D(x)}$ at the time point when this difference is maximized, using the trained $D(x)$. Right: $u_{\overline{D}}$ at the same time point using the constant diffusion coefficient. The constant value used for $D(x)$ corresponds to the minimum value of the trained diffusion coefficient, i.e., $\overline{D} = \min(D(x))$.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Definition 2.1
  • Proposition 2.3
  • Theorem 3.1
  • Proof 1
  • Theorem 3.2
  • Proof 2
  • Proposition 3.3
  • Proof 3
  • Remark 4.1
  • Lemma A.1: bungert2023uniform, Proposition 5.1
  • ...and 7 more