A Variational Mean-Field Control Framework for Graph Representation Learning

Abstract

Feature representation learning in graph neural networks (GNNs) is a dynamic process driven by progressive information exchange throughout the graph. Current GNNs typically apply pre-defined message-passing heuristics uniformly across all graph data, even when the assumed relational inductive bias conflicts with the intrinsic graph properties (e.g., heterophily). This limitation hinders a principled understanding of graph learning and motivates a shift from application-specific architectures toward a generalizable paradigm. In this work, we conceptualize GNN learning through the lens of mean-field control, where the desired learning outcome corresponds to a variational critical point, each node reaches a representation that is adaptively calibrated to both itself and the entire graph. Building on this formulation, we propose a mathematical framework based on mean-field control (MFC) to adaptively design GNNs for unseen graph data. Specifically, we jointly optimize two control patterns: diffusive mobility, governing information propagation across the graph, and reactive mobility, regulating feature transformation at individual nodes. Both controls are learned from each input instance by solving an MFC variational problem through Hamiltonian flows characterized by partial differential equations (PDE). Our variational framework unifies existing PDE-based GNNs as special mean-field games with fixed control patterns, and yields an end-to-end deep model, termed Nash-GNN. Extensive experiments show that Nash-GNN achieves state-of-the-art performance across diverse benchmarks, including heterophilic graphs and human connectomes. Beyond empirical gains, the MFC framework offers a principled lens to examine the mechanisms of graph representation learning through collective dynamics, opening promising directions for next-generation GNN design.

1. Introduction

Graph data is everywhere, from the social networks [1] to the complex brain connectomes that support cognition [2]. In this regard, graph neural networks (GNNs) that perform machine learning on graph data have been successfully deployed in various fields of science and engineering. Despite distinct instances of GNN model, the learning process typically consists of (1) a feature representation learning component for individual graph nodes and (2) a message passing component to characterize the interaction across graph nodes [3].

Meanwhile, the complexity of real-world graph data introduces new challenges. (1) Heterogeneity issue of graph data. Multiple lines of evidence show even simple multilayer perceptrons (MLPs) can outperform GNNs on heterophilic graphs [4], [5], [6], where the homophily assumption (i.e., strongly connected nodes are supposed to bear similar feature representations [7]) does not hold anymore. Tremendous efforts have been made to address this challenge, where representative approaches include extending single-channel message aggregation to adaptive channel mixing [8] and directly measuring the degree of graph heterophily [9]. (2) Over-smoothing issue. Excessive information aggregation between interconnected nodes causes graph feature representations to become similar after only a number of GNN layers. Many solutions have been proposed in the past several years. For example, graph neural diffusion (GRAND [10]) alleviates the over-smoothing issue by replacing the layer-by-layer optimization in the (discrete) GNN model with a continuous diffusion process, which is formulated in a partial differential equation (PDE). This approach has inspired the development of several new GNN models [11], [12], [13], [14], [15], [16], [17], [18] which are essentially derived from discretizations of different PDEs. Moreover, adjusting the network architecture provides an additional solution for the over-smoothing issue. For example, ResNet-based machine learning backbones have been integrated into GNNs [19], [20], alongside the adoption of Transformer backbone to efficiently capture global feature representations without the restriction of step-by-step traversal on the graph [21], [22].

Although tremendous efforts have been made to address the above challenges, the design of each GNN model is stereotyped upon the domain knowledge of machine learning and graph signal processing [23], [24]. Importantly, the design of GNN model is fixed for all graph data, and completely relies on back-propagation to optimize the learning performance by fine-tuning the model hyperparameters. Due to the lack of a good understanding of the learning mechanism inside GNNs at a system level, achieving state-of-the-art performance doesn’t necessarily ensure that the inference principle is well-suited for the underlying graph learning problem. A typical example is that a particular GNN model either yields superior performance on homophilic graphs compared to heterophilic ones, or vice versa. Thus, it is vital to unify the existing work of GNNs to an explainable learning paradigm (i.e., creating an explainable learning mechanism for GNNs) with well-defined principles and mathematically guaranteed foundations.

To establish a principled learning framework capable of generalizing to unseen graph data, we model the GNN learning process as a dynamical system of interacting particles. In this system, each graph node acts as a particle that optimizes its own feature representation while continuously exchanging information with its structural neighbors. This mechanism echoes recent works that conceptualize graph learning as a continuous process of heat diffusion [10], [12]. Theoretically, we treat the feature representations of nodes as their internal “potential energy” and hypothesize that the system’s dynamics adhere to the second law of thermodynamics [25]. Physically, a closed system evolves by dissipating its free energy to reach a stable equilibrium. According to Onsager’s reciprocal relations [25], this dissipative nature implies that “heat does not spontaneously flow from a colder body to a hotter body” [26]. Translated into the context of graph learning, this means that information inherently propagates from nodes with highly concentrated, discriminative features (hotter bodies) to adjacent nodes with less distinct features (colder bodies). Therefore, this natural diffusion process drives the entire network toward a robust, globally desired state.

Building on this perspective, we characterize the collective dynamics of simultaneous message passing and feature representation learning in GNNs through the lens of the second law of thermodynamics. In this formulation, the pattern of energy dissipation is governed by a learnable “free energy functional”. Drawing upon the mean-field theory in physics [27], we further conceptualize that the evolution of graph feature representations forms a graph-based mean-field density $\rho$. To structure this into a well-posed problem, we introduce the concept of mean-field control (MFC) [17] to examine the critical point of mean-field information dynamics. Thus, the most suitable relational inductive bias for a given graph is characterized by a continuous dynamic transport from the initial feature representations to a stable terminal state, which is governed by the downstream task. At this terminal state, each graph node attains a feature representation that harmonizes its localized intrinsic attributes with the global topological context of the entire graph.

Based on these physics principles, we formulate a novel variational MFC paradigm that is able to infer desired GNN architectures rather than relying on heuristic designs. From this viewpoint, the ideal relational inductive bias naturally materializes at the critical points of the mean-field dynamical system. The mathematical engine of our framework relies on learning two distinct macroscopic control parameters directly from the input data: a transport mobility function dictating how information propagates across the graph topology, and a reaction mobility function directing the internal feature transformation within individual nodes. By adjusting the dynamics trajectory of the system’s “free energy functional,” these dual mobility patterns can govern the network’s message-passing and representation-learning behaviors. Recognizing that each graph requires a tailored dynamic mechanism, our physics-guided architecture, coined as Nash-GNN¹, that operates through a bilevel hierarchical structure, encompassing both the abstract mechanism and the concrete model instance. The upper mechanism level is dedicated to fitting the MFC variational dynamics to the inherent characteristics of the target graph via the adaptive control patterns. These patterns then determine the Hamiltonian flows, which serve as the governing equations of the system. Due to the well-established mathematical equivalence between continuous PDEs and discrete GNNs [10], [11], [12], [28], the Hamiltonian flow operates as a bridge, translating the learned macroscopic control mechanisms into the localized fine-tuning of the network instance. Finally, Nash-GNN functions as an end-to-end deep learning pipeline that jointly discovers the tailored inductive bias and optimizes network parameters for any given graph.

The major contributions of this work are three-fold:

• A Unified variational principle: We introduce a physics-informed MFC framework that conceptualizes GNN learning as an adaptive reaction-diffusion process. This provides a mathematically foundation for discovering graph-specific learning mechanisms beyond heuristic designs.

• A unified theoretical framework: We demonstrate that our variational framework acts as a generalized mathematical foundation that unifies a wide spectrum of current PDE-based GNNs as distinct instances under specific mobility control patterns.

• Explainable neuroscience insights: Through extensive evaluations on both benchmarks and clinical human connectomes, we show that Nash-GNN not only achieves state-of-the-art performance but also offers a transparent lens to integrate complex neurodynamics mechanisms, such as the brain dynamics of Alzheimer’s and Parkinson’s diseases.

2. Methods

Consider an undirected, weighted graph $𝒢=(𝒱,ℬ)$, comprising a finite vertex set $𝒱={\left\{v_i\right\}}_{i=1}^N$ and an edge set denoted by $ℬ\subset 𝒱\times 𝒱$. The adjacency matrix is denoted as $\mathbf{A}={\left[a_{ij}\right]}_{i,j=1}^N$, where $a_{ij}>0$ iff $\left(v_i,v_j\right)\in ℬ$. Let $\mathbf{x}\left(t\right)=\left\{\mathbf{x}\right(v,t)\mid v\in 𝒱\}\in {\mathbb{R}}^{N\times d}$ represent the graph ensemble of node feature embeddings at a continuous layer time $t$, and $\mathbf{Y}\in \{1,\dots ,C{\}}^N$ denote the predefined labels over $C$ classes for downstream tasks.

Unlike traditional discrete GNNs that heuristically aggregate neighborhood information layer by layer, we conceptualize the feature representation learning on graphs as a continuous dynamic process governed by physical laws. Specifically, we cast this process into a generalized reaction-diffusion model (RDM):

$$\frac{\partial \mathbf{x}}{\partial t}=\mathrm{\Delta }F(\mathbf{x})+R(\mathbf{x})$$ (1)

where $\mathrm{\Delta }=\mathrm{d}\mathrm{i}\mathrm{v}(\nabla )$ denotes the graph Laplacian operator. Here, $F(\cdot )$ is the diffusion function that dictates how information propagates across the graph topology, and $R(\cdot )$ acts as the reaction function that regulates how individual node features transform independently of the graph structure.

Remark 1 (Physics Intuition of RDM):

Regard diffusion term $\mathrm{\Delta }F(\mathbf{x})$ as the “transportation network” that moves information between connected nodes (analogous to message passing), while the reaction term $R(\mathbf{x})$ acts as an “internal metabolism” transforming the features within each node itself (analogous to feature transformation via MLPs). By decoupling these two mechanisms, the RDM provides a holistic lens to investigate graph learning dynamics.

2.1. The Unification of GNNs via Reaction-Diffusion Dynamics

The beauty of the RDM framework lies in its capacity to serve as a unification of existing continuous and discrete GNN architectures. By imposing specific constraints on the diffusion $F(\mathbf{x})$ and reaction $R(\mathbf{x})$ functions, current state-of-the-art models inherently degenerate into special cases of our RDM paradigm:

GRAND [10].

Graph neural diffusion models the message passing as a pure heat diffusion process, governed by $\frac{\partial \mathbf{x}(t)}{\partial t}=\mathrm{d}\mathrm{i}\mathrm{v}[c\nabla \mathbf{x}(t)]$. Assuming a homogeneous and time-invariant diffusion scalar $c$, this translates to our RDM framework by setting $F(\mathbf{x})=\mathbf{x}$ and $R(\mathbf{x})=0$.

GraphBel [29].

Extended from GRAND, GraphBel employs Beltrami flow to normalize the graph gradient, yielding $\frac{\partial \mathbf{x}(t)}{\partial t}=\Vert \nabla \mathbf{x}{\Vert }^{-1}\mathrm{\Delta }\mathbf{x}(t)$. Here, the diffusion remains structurally linear $F(\mathbf{x})=\mathbf{x}$ with $R(\mathbf{x})=0$, scaled by the inverse gradient magnitude.

ACMP [11].

By using the principles of interacting particle dynamics, the Allen-Cahn message passing network formulates attractive and repulsive forces through a double-well potential $f(\mathbf{x})$. Its core PDE simplifies to $\frac{\partial \mathbf{x}(t)}{\partial t}=\mathrm{\Delta }\mathbf{x}(t)-f^{\prime }(\mathbf{x})$, which aligns with the RDM by setting $F(\mathbf{x})=\mathbf{x}$ and activating the reaction term as $R(\mathbf{x})=-f^{\prime }(\mathbf{x})$.

GraphCON [13].

Inspired by the nonlinear Klein-Gordon equation, GraphCON introduces a second-order ODE with a damping term $\gamma$ to mitigate over-smoothing. Its dynamics can be viewed as an extension of RDM where the reaction term $R(\mathbf{x})$ incorporates both a nonlinear potential and a velocity-dependent damping $-\gamma \dot{\mathbf{x}}$. By setting $F(\mathbf{x})=\mathbf{x}$ and $R(\mathbf{x})=\sigma (\mathbf{A}\mathbf{x}\mathbf{W})-\gamma \dot{\mathbf{x}}$, GraphCON becomes a kinetic variant of our RDM framework.

PairNorm [30].

PairNorm aims to maintain the total variance of node features during diffusion to prevent oversmoothing. In the continuous limit, this dynamics corresponds to an RDM where the diffusion is driven by standard message passing, while the reaction term enforces a geometric normalization penalty. Here, $F(\mathbf{x})=\mathbf{x}$ and $R(\mathbf{x})=\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}(\mathbf{x})-\mathbf{x}$, where the operator $\mathrm{N}\mathrm{o}\mathrm{r}\mathrm{m}(\cdot )$ centers and scales the features to ensure $\sum _i{\mathbf{x}}_i=\mathbf{0}$ and $\frac{1}{N}\sum _i{‖{\mathbf{x}}_i‖}^2=s$.

SHEAF [31].

This model generalizes the standard graph Laplacian to a sheaf Laplacian, where the diffusion is governed by local restriction maps (matrices) on edges rather than scalar weights. Under our RDM umbrella, this represents a sophisticated transition from isotropic diffusion to anisotropic transport. Here, $F(\mathbf{x})=\mathbf{V}\mathbf{x}$, where $\mathbf{V}$ denotes the fiber-wise linear maps defined over the tangent bundle of the graph, and setting the reaction term to $R(\mathbf{x})=\mathbf{0}$ (for pure sheaf diffusion without additional local forcing).

The Limitation of Current Paradigms:

While existing PDE-based GNNs demonstrate empirical success, they rigidly apply a pre-defined RDM mechanism (fixed $F$ and $R$) to all graph data. This “one-size-fits-all” approach lacks a system-level adaptability. Different graphs (e.g., homophilic vs. heterophilic) require vastly different balances between diffusion and reaction. This observation naturally begs the question: Instead of hand-crafting these functions, can we mathematically formulate an adaptive RDM mechanism tailored to the specific properties of a given graph?

2.2. A Variational Framework of Mean-Field Control for Exploring Learning Mechanism in GNNs

To answer the aforementioned question, we elevate the perspective from defining deterministic PDEs to solving an optimal control problem. We propose a variational MFC framework to derive an adaptive learning mechanism (i.e., a graph-tailored inductive bias) for GNNs.

2.2.1. Problem Formulation: From Particles to Mean-Field

As illustrated in Fig. 1, instead of tracking individual node features deterministically, we treat the ensemble of graph embeddings $\mathbf{x}(t)$ as a continuous density distribution, named the mean-field density $\rho (t)$. Each node acts as an interacting particle within this field. The evolution of this density over time satisfies the continuity equation:

$${\partial }_t\rho (t)=-\mathrm{d}\mathrm{i}\mathrm{v}(\rho (t)\mathbf{z}(t))$$ (2)

where $\mathbf{z}(t)$ denotes the latent velocity field driving the feature updates.

Fig. 1.

Overview of MFC framework for GNNs. Consider the distribution of initial graph representation as a mean-field density ${\rho }_0$ on graph $𝒢$. The learning process in GNN can be formulated as finding a potential function $\mathrm{\Phi }$ that drives the density evolution until the terminal distribution ${\rho }_T$ is reached. Furthermore, we propose to learn the control patterns from the underlying graph data, which allows us to customize the optimal transport from ${\rho }_0$ to ${\rho }_T$.

By constraining the evolution of particles $\mathbf{x}(t)$ to be a gradient flow that minimizes a certain global energy functional $ℰ\left({\rho }_t\right)={\int }_{𝒢}U\left({\rho }_t\right)dv$, mathematical optimal transport theory guarantees the existence of a unique potential function ${\mathrm{\Phi }}_t$ such that ${\mathbf{z}}_t=\nabla {\mathrm{\Phi }}_t$.

Remark 2 (Physics Intuition of Mean-Field Control):

Imagine training a GNN as guiding a flock of birds (nodes) from random initial positions ${\rho }_0$ to a structured formation ${\rho }_T$ that accurately predicts the downstream task. Instead of manually giving instructions to each bird (standard message passing), mean-field control designs an invisible “energy landscape” (the potential function $\mathrm{\Phi }$). The birds naturally follow the gradient of this landscape $(\nabla \mathrm{\Phi })$ to reach a stable collective destination smoothly and efficiently.

Within this formulation, the raw node features serve as the initial state ${\rho }_0$ at $t=0$, and the desired feature representations for downstream tasks (e.g., node classification) define the terminal state ${\rho }_T$ at $t=T$. The learning process of GNNs is fundamentally transformed into an optimal transport problem: What is the most energy-efficient dynamics (i.e., task-specific $F$ and $R$) to transport ${\rho }_0$ to ${\rho }_T$? Studying the critical points of this energy functional $ℰ(\rho )$ is able to lead us to the mean-field control problem.

2.2.2. Translating RDM into a Mean-Field Control Problem

Instead of heuristically imposing predefined diffusion and reaction terms, we can systematically derive the governing dynamics by minimizing a global objective functional. Let $\rho (t,v)$ denote the continuous state distribution (i.e., the mean-field density) of node features at layer time $t$. We construct a free energy functional:

$$ℰ\left(\rho \right)={\int }_{𝒢}U\left(\rho \left(t,v\right)\right)dv,$$ (3)

where $U:\mathbb{R}\to \mathbb{R}$ is a strictly convex internal energy density function with $U^{\prime \prime }>0$.

Assuming the evolution of the feature density $\rho (t,v)$ follows the generalized RDM defined in Eq. 1, the dissipation of this free energy can be rigorously formulated as a gradient flow on a specific probability manifold:

$$\frac{\partial }{\partial t}{\rho }_t=-{\mathbf{G}}_{\rho }^{-1}\frac{\delta ℰ\left({\rho }_t\right)}{\delta {\rho }_t},$$ (4)

$${\mathbf{G}}_{\rho }^{-1}(\psi )=-\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{F^{\prime }(\rho )}{U^{\prime \prime }(\rho )}\nabla \psi \right)-\frac{R(\rho )}{U^{\prime }(\rho )}\psi ,$$ (5)

where $\frac{\delta ℰ}{\delta \rho }=U^{\prime }(\rho )$ is the first variation (Fréchet derivative), $\psi$ is a generic test function on which the operator ${\mathbf{G}}_{\rho }^{-1}$ acts, and ${\mathbf{G}}_{\rho }$ acts as a positive-definite metric tensor operator characterizing the underlying geometry [32].

Proof. To demonstrate that the macroscopic RDM dynamics naturally emerge from this gradient flow, we compute the right-hand side of Eq. 4. Substituting the Fréchet derivative $\frac{\delta ℰ}{\delta \rho }=U^{\prime }(\rho )$ into the gradient flow alongside the metric tensor ${\mathbf{G}}_{\rho }$, we obtain:

$$\frac{\partial \rho }{\partial t}=\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{F^{\prime }(\rho )}{U^{\prime \prime }(\rho )}\nabla U^{\prime }(\rho )\right)+\frac{R(\rho )}{U^{\prime }(\rho )}{U}^{\prime }(\rho ).$$

Applying the chain rule to the spatial gradient of the energy variation yields $\nabla U^{\prime }(\rho )=U^{\prime \prime }(\rho )\nabla \rho$. Substituting this into the diffusion term strictly cancels out the $U^{\prime \prime }(\rho )$ denominator:

$$\mathrm{d}\mathrm{i}\mathrm{v}\left(\frac{F^{\prime }(\rho )}{U^{\prime \prime }(\rho )}{U}^{\prime \prime }(\rho )\nabla \rho \right)=\mathrm{d}\mathrm{i}\mathrm{v}\left(F^{\prime }(\rho )\nabla \rho \right).$$

By the chain rule again, $\nabla F(\rho )=F^{\prime }(\rho )\nabla \rho$. Thus, the diffusion term simplifies perfectly to the graph Laplacian of the diffusion function $\mathrm{\Delta }F(\rho )$. Simultaneously, the $U^{\prime }(\rho )$ cancels out in the reaction term. Combining both, the gradient flow meticulously recovers $\frac{\partial \rho }{\partial t}=\mathrm{\Delta }F(\rho )+R(\rho )$. ◼

To bridge this dynamical system with our proposed mean-field control framework, we package the complex fractional terms in the metric tensor into two distinct, learnable mobility functions.

Definition 1 (Mobility Control Patterns).

Given the diffusion function $F$, the reaction function $R$, and the energy density $U$, we define the diffusive mobility ${\eta }_1(\rho )=\frac{F^{\prime }(\rho )}{U^{\prime \prime }(\rho )}$ and the reactive mobility. ${\eta }_2(\rho )=-\frac{R(\rho )}{U^{\prime }(\rho )}$.

Remark 3 (Bridging PDEs and Control Mechanisms):

By substituting these mobility terms, the interaction between the objective functional $ℰ(\rho )$ and the generalized RDM is gracefully disentangled. This substitution reveals a key insight: the specific behaviors of different GNN models are fundamentally determined by their underlying mobility patterns ${\eta }_1$ and ${\eta }_2$. Rather than manually searching for the desired $F$ and $R$ within the vast, intractable PDE space, our formulation translates the graph learning objective into learning the optimal control patterns ${\eta }_1$ and ${\eta }_2$. This provides a mathematically tractable and principled path for the subsequent mean-field control optimization.

2.2.3. Deriving Learning Principles via Hamiltonian Mechanics

Suppose ${\rho }_0$ represents the initial distribution of graph features. Given the specific mobility patterns ${\eta }_1$ and ${\eta }_2$, optimal transport theory reveals that the state evolution from ${\rho }_0$ to a terminal distribution ${\rho }_T$ is governed by a coupled potential function $\mathrm{\Phi }$. By analyzing the critical points of the mean-field control problem, we can naturally derive the corresponding Hamiltonian equations:

$$\left\{\begin{array}{c}{\partial }_t\rho +\mathrm{d}\mathrm{i}\mathrm{v}(\rho \nabla \mathrm{\Phi })=0\\ {\partial }_t\mathrm{\Phi }+\mathscr{H}(\rho ,\nabla \mathrm{\Phi })=0\end{array}\right.$$ (6)

Herein, $\mathscr{H}(\rho ,\nabla \mathrm{\Phi })$ is the Hamiltonian density induced by the mobilities $\left({\eta }_1,{\eta }_2\right)$. Equivalently, the system is generated by the Hamiltonian functional ${ℱ}_h(\rho ,\mathrm{\Phi })={\int }_{𝒢}\mathscr{H}(\rho ,\nabla \mathrm{\Phi })dv$, whose canonical form will be formally introduced in Section 2.3.2.

In this dynamical system, the Hamiltonian flow $\nabla \mathrm{\Phi }$ (illustrated by the blue arrows in Fig. 1) determines the trajectory of feature updates for each interacting node. Thus, the relational inductive bias of a GNN, traditionally handcrafted through discrete heuristic message-passing layers, can now be understood as an adaptive process conforming to this Hamiltonian mechanics.

Rather than assuming a universal, fixed physical law for all graph data, our framework posits that an adaptive learning mechanism is intrinsically graph-dependent. It is characterized by the target control patterns ${\eta }_1$ and ${\eta }_2$, which are optimized to ensure that the terminal state ${\rho }_T$ aligns with the specific downstream objectives (e.g., node classification), leveraging PDE-based solvers [28].

To operationalize this principle, we instantiate the framework as the two-level pipeline illustrated in Fig. 2. At the macroscopic level, variational analysis of the RDM translates graph learning into an MFC problem, whose critical point yields the mobility control patterns $\left({\eta }_1,{\eta }_2\right)$. At the microscopic level, these mobilities define the Hamiltonian flow (Eq. 6), which is integrated to evolve node representations toward the terminal state ${\rho }_T$. Jointly optimizing both levels end-to-end produces Nash-GNN, whose concrete neural architecture is detailed next in Section 2.3.

Fig. 2.

The methodology scheme of our MFC framework. We first recast conventional GNNs as a continuous reaction-diffusion model (RDM), and then apply variational analysis within mean-field theory to formulate graph representation learning as a mean-field control (MFC) problem. The resulting MFC problem is characterized by two mobility control patterns: the diffusive mobility ${\eta }_1$ and the reactive mobility ${\eta }_2$, which jointly govern how information is propagated and how individual node features are transformed. Studying the critical points of this variational problem yields the governing Hamiltonian flow, which is numerically integrated to produce Nash-GNN, an end-to-end neural instantiation of the proposed pipeline.

2.3. Operationalizing the Framework: The Nash-GNN Architecture

Building upon the theoretical foundation of mean-field control, we now present Nash-GNN, an end-to-end neural architecture that practically instantiates the proposed variational framework.

2.3.1. Revisiting Existing GNNs under the New Lens

Before delving into the specific neural design, it is crucial to clarify how our framework encapsulates existing methodologies. As summarized in Table 1, classical PDE-based GNNs can be subsumed as specific instances of our framework with fixed internal energy functionals $U(\rho )$ and mobility patterns. For instance, the diversity in their learning behaviors stems directly from their isolated choices of diffusion and reaction dynamics. In contrast, Nash-GNN does not rely on these handcrafted heuristics, instead, it parameterizes and learns these underlying control variables $\left({\eta }_1,{\eta }_2\right)$ from data, allowing the model to smoothly transition between different dynamic regimes depending on the graph’s heterophily or homophily.

TABLE 1.

Unification of existing GNN models: Variational functionals $ℰ(\rho )={\int }_{𝒢}U(\rho )dv$, diffusion function $F(\cdot )$, reaction function $R(\cdot )$, mobility functions ${\eta }_1(\cdot )$ and ${\eta }_2(\cdot )$, alongside their resulting Hamiltonian equations. Here, ${\mathbf{L}}_{ℱ}$ denotes the Sheaf Laplacian tensor, $\bar{\rho }$ represents the mean feature density, and $f_{\theta }(\cdot )$ denotes a non-linear neural transformation.

Model	$ℰ(\rho )={\int }_{𝒢}U(\rho )dv$	$F(\rho )$	$R(\rho )$	${\eta }_1(\rho )=\frac{F^{\prime }}{U^{\prime \prime }}$	${\eta }_2(\rho )=-\frac{R}{U^{\prime }}$	Hamiltonian Equation
GRAND	${\int }_{𝒢}(\rho \mathrm{l}\mathrm{o}\mathrm{g}\rho -1)dv$	$\rho$	0	$\rho$	0	$\left\{\begin{array}{l}{\partial }_t\rho +\nabla \cdot (\rho \nabla \mathrm{\Phi })=0\\ {\partial }_t\mathrm{\Phi }+\frac{1}{2}\Vert \nabla \mathrm{\Phi }{\Vert }^2=0\end{array}\right.$
GraphBel	${\int }_{𝒢}\left(\frac{1}{2}{\rho }^2\right)dv$	$\rho$	0	1	0	$\left\{\begin{array}{l}{\partial }_t\rho +\nabla \cdot (\nabla \mathrm{\Phi })=0\\ {\partial }_t\mathrm{\Phi }=0\end{array}\right.$
ACMP	${\int }_{𝒢}f(\rho )dv$	$\rho$	$-f^{\prime }(\rho )$	$\frac{1}{f^{\prime \prime }(\rho )}$	1	$\left\{\begin{array}{l}{\partial }_t\rho +\nabla \cdot \left(f^{\prime \prime }(\rho {)}^{-1}\nabla \mathrm{\Phi }\right)-\mathrm{\Phi }=0\\ {\partial }_t\mathrm{\Phi }-\frac{1}{2}\Vert \nabla \mathrm{\Phi }{\Vert }^2\frac{f^{\prime \prime \prime }(\rho )}{f^{\prime \prime }(\rho {)}^2}=0\end{array}\right.$
GraphCON	${\int }_{𝒢}\left(\frac{1}{2}{\rho }^2\right)dv$	$\rho$	$f_{\theta }(\rho )-\gamma {\partial }_t\rho$	1	$\frac{\gamma {\partial }_t\rho -f_{\theta }(\rho )}{\rho }$	$\left\{\begin{array}{l}{\partial }_t\rho +\nabla \cdot (\nabla \mathrm{\Phi })+{\eta }_2\rho =0\\ {\partial }_t\mathrm{\Phi }-{\mathscr{H}}_{con}(\rho ,\nabla \mathrm{\Phi })=0\end{array}\right.$
PairNorm	${\int }_{𝒢}\left(\frac{1}{2}{\rho }^2\right)dv$	$\rho$	$-\lambda (\rho -\bar{\rho })$	1	$\lambda \left(1-\frac{\bar{\rho }}{\rho }\right)$	$\left\{\begin{array}{l}{\partial }_t\rho +\nabla \cdot (\nabla \mathrm{\Phi })+{\eta }_2\rho =0\\ {\partial }_t\mathrm{\Phi }-{\mathscr{H}}_{\mathit{norm}}(\rho ,\nabla \mathrm{\Phi })=0\end{array}\right.$
SHEAF	${\int }_{𝒢}\left(\frac{1}{2}{\rho }^{\mathrm{\top }}{\mathbf{L}}_{ℱ}\rho \right)dv$	${\mathbf{L}}_{ℱ}\rho$	0	${\mathbf{L}}_{ℱ}$	0	$\left\{\begin{array}{l}{\partial }_t\rho +\nabla \cdot \left({\mathbf{L}}_{ℱ}\nabla \mathrm{\Phi }\right)=0\\ {\partial }_t\mathrm{\Phi }+\frac{1}{2}{‖{\mathbf{L}}_{ℱ}\nabla \mathrm{\Phi }‖}^2=0\end{array}\right.$

2.3.2. Network Architecture of Nash-GNN

As illustrated in Fig. 3, the architecture of Nash-GNN comprises two intertwined neural components: a macroscopic adaptive controller ${ℱ}_m$ that parameterizes the optimal control patterns, and a microscopic Hamiltonian functional ${ℱ}_h$ that induces the Hamiltonian flow governing the evolution of node representations.

Fig. 3.

The network architecture of Nash-GNN. The architecture comprises two intertwined neural components: (1) the macroscopic adaptive controller ${ℱ}_m$ and (2) the microscopic Hamiltonian functional ${ℱ}_h$. Specifically, we employ an input convex neural network (ICNN) [33] to parameterize the mobility control patterns ${\eta }_1$ and ${\eta }_2$, which strictly shape the characteristics of the Hamiltonian flow governing the reaction-diffusion dynamics. Given ${\eta }_1$ and ${\eta }_2$, the Hamiltonian functional ${ℱ}_h(\rho ,\mathrm{\Phi })={\int }_{𝒢}\mathscr{H}(\rho ,\nabla \mathrm{\Phi })dv$ induces the flow ${\partial }_t\rho =\delta {ℱ}_h/\delta \mathrm{\Phi }$ and ${\partial }_t\mathrm{\Phi }=-\delta {ℱ}_h/\delta \rho$, which is numerically integrated by a standard PDE solver (Euler in our experiments) to continuously evolve the potential function $\mathrm{\Phi }$ and the mean-field density $\rho$. The entire system is driven end-to-end such that the terminal density ${\rho }_T$ yields the desired feature representations for downstream graph learning tasks.

Macroscopic adaptive controller ${ℱ}_m$: Parameterizing mobility.

The evolution of the mean-field density is uniquely determined by the mobility functions ${\eta }_1$ and ${\eta }_2$. To ensure the well-posedness of the corresponding Hamiltonian system, the underlying internal energy functional $U(\rho )$ must maintain strict convexity. To enforce this mathematical prerequisite within a deep learning framework, we parameterize the mapping from the current phase space $\left({\rho }_t,{\mathrm{\Phi }}_t\right)$ to the mobilities $\left({\eta }_1,{\eta }_2\right)$ using an input convex neural network (ICNN) [33].

The ICNN guarantees convexity with respect to its inputs by restricting the weight matrices of the hidden layers to be non-negative. Specifically, the network processes the phase space through recursive layers:

$${\xi }^{(k+1)}={\sigma }^{(k)}\left({\mathbf{W}}_1^{(k)}{\xi }^{(k)}+{\mathbf{W}}_2^{(k)}\left[{\rho }_t\Vert {\mathrm{\Phi }}_t\right]+{\mathbf{b}}^{(k)}\right),$$ (7)

where $\left[{\rho }_t\Vert {\mathrm{\Phi }}_t\right]$ denotes the concatenation of the feature density and the potential function at continuous time $t$. Here, ${\mathbf{W}}_1^{(k)}\ge 0$ ensures the preservation of convexity, and ${\sigma }^{(k)}$ is a convex and monotonically non-decreasing activation function. Through this mechanism, ${ℱ}_m$ acts as an intelligent “oracle”, dynamically suggesting the graphspecific diffusion and reaction rates based on the instantaneous topological and feature states.

Microscopic Hamiltonian functional ${ℱ}_h$: Inducing Hamiltonian flows.

Given the generated mobility patterns ${\eta }_1$ and ${\eta }_2$, the microscopic node feature evolution strictly adheres to the Hamiltonian dynamics defined in Eq. 6. We conceptualize the graph feature representations $\rho$ and the latent potential function $\mathrm{\Phi }$ as the generalized position and momentum variables, respectively, within the phase space. Introducing the Hamiltonian functional

$${ℱ}_h(\rho ,\mathrm{\Phi })={\int }_{𝒢}\mathscr{H}(\rho ,\nabla \mathrm{\Phi })dv,$$ (8)

the system in Eq. 6 can be equivalently written in canonical form,

$${\partial }_t\rho =\frac{\delta {ℱ}_h}{\delta \mathrm{\Phi }},{\partial }_t\mathrm{\Phi }=-\frac{\delta {ℱ}_h}{\delta \rho },$$ (9)

whose specific form is determined by the learned mobilities $\left({\eta }_1,{\eta }_2\right)$ (see Table 1 for concrete instances). The resulting flow is realized numerically by a standard PDE solver (Euler in our experiments).

The forward pass of Nash-GNN is formulated as an integration problem over a continuous layer-time domain $t\in [0,T]$. Initially, a set of multi-layer perceptrons (MLPs), denoted as ${ℱ}_p$, projects the raw node features x into the initial phase space $\left({\rho }_0,{\mathrm{\Phi }}_0\right)$. Driven by the Hamiltonian flow induced by ${ℱ}_h$, the state evolves continuously:

$$\left({\rho }_T,{\mathrm{\Phi }}_T\right)=\left({\rho }_0,{\mathrm{\Phi }}_0\right)+{\int }_0^T\left[\begin{array}{c}\delta {ℱ}_h/\delta \mathrm{\Phi }\\ -\delta {ℱ}_h/\delta \rho \end{array}\right]dt.$$ (10)

In practice, this integration is executed using numerical PDE solvers (e.g., Runge–Kutta methods). At the terminal time $T$, a canonical projection function ${ℱ}_{\mathrm{\Pi }}$ extracts the evolved density ${\rho }_T$ from the phase space to yield the final representations ${\mathbf{x}}^{(T)}={ℱ}_{\mathrm{\Pi }}\left({\rho }_T,{\mathrm{\Phi }}_T\right)$. These representations are intrinsically optimized for the specific graph topology and are subsequently fed into a task-specific classifier (e.g., cross-entropy loss for node classification) to enable end-to-end backpropagation.

Potential of MFC framework in the realm of GNN.

Our MFC framework for the design principle of GNN models offers a set of new insights that might benefit the GNN community.

Enhanced model explainability.

The well-studied principles in mean-field theory and information geometry [34] enable the formulation of graph data learning as a variational problem within a metric space $(ℳ,\mathbf{G})$. Specifically, we introduce the concept of the control patterns associated with the gradient flow, which allows us to establish a highlevel understanding of the behavior of GNN model. In Section 3.2, we demonstrate new insights into graph learning that are beyond conventional graph attention approaches.

A mathematical framework linking continuous formulation and discrete model.

Inspired by the recent PDE work (such as Neural ODE and GRAND), we conceptualize the layer-by-layer process of graph representation learning as a dynamical system in a continuous domain. Furthermore, our MFC framework extends the diffusion process to a scenario of the reaction-diffusion model, where the system dynamics follow the second law of thermodynamics. To that end, we establish a connection between the learning mechanism in the continuous domain, which offers clear explainability, and the discrete GNN model, which harnesses the power of machine learning. The conjunction is that we can design an effective GNN model for the unseen graph data by learning the most appropriate mechanism of how feature representations are projected and propagated throughout the graph, which translates heuristic model design into a mathematically principled, adaptive control problem.

An efficient approach to alleviate the over-smoothing issue in GNNs.

The learned control patterns are used to effectively align the underlying message-passing mechanism with the characteristics of the graph topology by promoting information exchange in homophilic graphs while restraining the diffusion process in heterophilic graphs. To that end, our MFC framework provides a selective mechanism to avoid excessive information exchange, which allows us to maintain the discrimination power of graph feature representations in a very deep network architecture.

3. EXPERIMENTS

Datasets.

The empirical validation of Nash-GNN has a dual purpose. Beyond merely benchmarking its performance against state-of-the-art GNN architectures, it is designed as an in-depth mechanistic investigation to elucidate the underlying dynamics of graph learning. Specifically, the evaluations include:

• Graph node classification: We evaluate on both heterophilic and homophilic graphs (sorted by homophily ratio $h$ [4]): Texas $(h=0.11)$, Wisconsin $(h=0.21)$, Actor $(h=0.22)$, Squirrel $(h=0.22)$, Chameleon $(h=0.23)$, Cornell $(h=0.3)$, Citeseer $(h=0.74)$, Pubmed $(h=0.8)$, and Cora $(h=0.81)$. Furthermore, we verify the scalability of Nash-GNN on large-scale datasets from the Open Graph Benchmark (OGB) [35]: ogbn-arxiv and ogbn-products. Detailed statistics are provided in Table 2.

• Graph classification: First, we conduct experiments on human connectomes for disease diagnosis using processed neuroimaging data from ABIDE (Autism) [36], ADNI (Alzheimer’s) [37], OASIS (Alzheimer’s) [38], and PPMI (Parkinson’s) [39]. We utilize regional BOLD (blood oxygenation level-dependent) time series as node embeddings and functional connectivity (FC) using the AAL atlas (116 regions) [40] and Destrieux atlas (160 regions) [41] as adjacency matrices. For the ADNI dataset, subjects are categorized into cognitive normal (CN), subjective memory concern (SMC), early-stage mild cognitive impairment (EMCI), late-stage mild cognitive impairment (LMCI), and Alzheimer’s Disease (AD) groups. To facilitate binary evaluation, we group CN, SMC, and EMCI as the “CN-like” cohort, while LMCI and AD form the “AD-like” cohort [42]. The PPMI dataset consists of four distinct clinical stages: normal control, scans without evidence of dopaminergic deficit (SWEDD), prodromal, and Parkinson’s disease (PD). Second, to demonstrate broad generalizability, we benchmark on the TUDataset [43], including MUTAG, NCI1, ENZYMES, DD, PTC_FM, IMDB-B, and PROTEINS. Details are summarized in Table 3.

TABLE 2.

Data description on node classification.

	Texas	Wisconsin	Actor	Squirrel	Chameleon	Cornell	Citeseer	Pubmed	Cora	ogbn-arxiv	ogbn-products
Hom. ratio $h$	0.11	0.21	0.22	0.22	0.23	0.3	0.57	0.74	0.81	0.23	0.41
#Nodes $\left|𝒱\right|$	183	251	7,600	5,201	2,277	183	3,327	19,717	2,708	169,343	2,449,029
#Edges |$𝒫$|	295	466	26,752	198,493	31,421	280	4,676	44,327	5,278	1,116,243	61,859,140
#Classes |$𝒴|$	5	5	5	5	5	5	7	3	6	23	27

TABLE 3.

Data description on graph classification.

	ABIDE	ADNI	OASIS	PPMI	MUTAGE	NCI1	ENZYMES	D&D	PTC_FM	IMDB	PROTEINS
Condition	Autism	Alzheimer	Alzheimer	Parkinson	-	-	-	-	-	-	-
Avg #Nodes $\left|𝒱\right|$	116	116	160	116	17.89	29.87	32.63	284.32	14.11	19.77	39.36
Avg #Edges $\left|𝒫\right|$	13,456	13,456	25,600	13,456	19.79	32.30	62.14	715.66	14.48	19.77	39.06
#Classes |$𝒴|$	2	5	2	4	2	2	6	2	2	2	2
#Graphs $\left|𝒢\right|$	1,025	250	1,475	209	188	4,110	600	1,178	349	1,000	1,113

Experiment Setup.

We compare Nash-GNN against representative GNN baselines, including vanilla GCN [7], GAT [44], GraphSAGE [45], GraphCON [13], GraphBel [29], GRAND [10], ACMP [11], HANG [28], GCNII [46] and AM-GCN [47]. For node classification on standard benchmarks, we adopt the widely-used data-splitting protocol (60% training, 10% validation, 30% testing) [48]. For the TUDataset and medical connectome data, we report 5-fold cross-validation results. Models are optimized using the Adam optimizer with a learning rate of 0.01 over 250 epochs. Hidden dimensions are intrinsically set to 128 (with 64 for Squirrel/Chameleon and 32 for Cora/ABIDE to prevent overfitting). All experiments are executed on a workstation equipped with four NVIDIA RTX 6000 Ada GPUs. The specific details of the algorithm are as provided in Algorithms 1 and 2.

3.1. Benchmark Evaluations: Validating the Mean-Field Control Dynamics

3.1.1. Performance on Graph Node Classification

Results.

Table 4 shows the quantitative comparisons across nine standard datasets, and Table 5 presents the evaluation on the large-scale OGB datasets. Nash-GNN consistently establishes state-of-the-art performance across both heterophilic and homophilic topologies, demonstrating significant superiority over existing handcrafted GNN models.

TABLE 4.

Test accuracies (%) on nine graph networks for node classification task.

Dataset	GCN	GAT	GraphSAGE	GraphCON	GraphBel	GRAND	ACMP	HANG	Nash-GNN
Texas	61.48±7.68	61.48±5.61	81.85±5.46	84.20±3.01	85.71±2.96	85.52±0.57	86.00±3.21	85.56±1.54	94.00*±2.21
Wisconsin	55.00±5.79	49.86±3.47	81.07±3.82	87.93±4.30	86.95±3.53	87.35±5.37	86.49±4.30	84.27±4.75	88.22*±4.16
Actor	29.05±1.69	26.34±0.87	32.72±3.62	35.62±1.54	33.58±1.61	33.20±1.25	33.65±1.95	35.12±1.06	36.25*±1.38
Squirrel	39.27±1.09	36.05±1.69	39.76±1.88	34.90±1.90	39.08±6.49	35.46±1.01	36.56±1.52	39.32±1.28	40.38*±0.60
Chameleon	55.65±3.54	51.57±4.93	55.12±1.84	48.31±2.77	46.79±1.61	46.03±1.89	46.85±1.95	59.85±2.12	60.14±0.22
Cornell	50.00±7.25	40.37±4.61	67.78±7.81	82.16±6.17	78.53±1.28	80.00±2.31	79.73±2.86	77.41±4.01	83.59*±2.23
Citeseer	66.51±12.75	46.13±19.72	63.16±9.34	74.84±0.49	69.62±0.56	74.98±1.45	75.07±2.17	73.75±1.80	77.61*±1.38
Pubmed	88.63±1.31	87.36±1.34	88.71±0.34	88.78±0.46	86.97±0.37	88.44±0.34	87.76±1.24	89.93±0.27	90.75±0.27
Cora	86.86±0.63	87.04±0.69	81.38±6.24	86.27±0.51	82.60±0.64	87.53±0.59	82.91±2.62	84.56±1.21	87.80±0.47

Statistical significance is assessed based on 20 resampling tests conducted using a randomized seed. ‘*’ means statistically significance with p ≤ 0.05.

TABLE 5.

The performance on OGB dataset.

	GCN	GAT	GraphSAGE	GraphBel	ACMP	HANG	GCNII	AM-GCN	Nash-GNN
ogbn-arxiv	0.7174	0.7365	0.7149	0.7256	0.7543	0.7484	0.7274	0.7239	0.7697
ogbn-products	0.7564	0.7904	0.7870	0.8049	0.8295	0.8468	0.7824	0.8013	0.8740

Discussion.

These empirical results highlight the theoretical efficacy of our variational framework. Instead of employing a rigid, “one-size-fits-all” message-passing scheme, Nash-GNN autonomously customizes the continuous control mechanisms to adapt to the specific properties of each graph. Importantly, this success validates our core hypothesis: graph representation learning is fundamentally a mean-field control process governed by adaptive Hamiltonian mechanics, rather than a heuristic aggregation procedure.

3.1.2. Ablation Study

To better understand the architectural advantages of Nash-GNN, we conduct ablation studies focusing on the parameterization of the Hamiltonian system and the selection of PDE solvers. Numerical results are provided in Table 6.

TABLE 6.

Ablation studies on Hamiltonian energy function $\mathscr{H}$ and PDE solver.

	Texas	Wisconsin	Actor	Squirrel	Chameleon	Cornell	Citeseer	Pubmed	Cora
Hamiltonian energy function $\mathscr{H}$
GCN	0.8715	0.8200	0.3403	0.4019	0.5522	0.7495	0.7075	0.8880	0.8739
GAT	0.9083	0.8466	0.3544	0.3506	0.5913	0.7266	0.7313	0.8895	0.8703
PDE Solver
Euler	0.9266	0.8733	0.3529	0.4013	0.6043	0.8716	0.7663	0.9037	0.8832
RK4	0.8073	0.8700	0.3912	0.4415	0.6565	0.8807	0.76627	0.8896	0.8867
Symplectic-Euler	0.8716	0.8600	0.3586	0.4013	0.6130	0.8165	0.7155	0.8769	0.8686
dopri5	0.9541	0.8767	0.3917	0.3795	0.6522	0.8624	0.7394	0.8911	0.8467

Regarding Hamiltonian energy parameterization:

In classical physical systems, energy is often characterized explicitly by pairwise interactions (e.g., nodes connected by springs of given stiffness [49]). Translating this intuition to GNNs, one might attempt to manually pre-define explicit Hamiltonians $\mathscr{H}$ based on local neighborhood aggregations (conceptually akin to the functional forms driving GCN or GAT). However, as evidenced in Table 6, such heuristic simplification severely restricts the model’s adaptability across diverse graph properties, leading to degraded performance. In contrast, Nash-GNN circumvents the intractable task of hand-crafting explicit Hamiltonians. Instead, it learns the underlying Hamiltonian $\mathscr{H}$ by parameterizing the mobility control patterns $\left({\eta }_1,{\eta }_2\right)$ via the macroscopic adaptive controller ${ℱ}_m$. The deployment of an ICNN [33] to generate these mobilities is not an empirical trick, but a strict mathematical necessity: it guarantees the convexity of the implicit energy landscape, thereby ensuring the well-posedness and stability of the entire Hamiltonian flow.

Regarding numerical PDE solvers:

The continuous nature of our Hamiltonian flow necessitates a numerical integrator. We evaluated various forward ODE/PDE solvers, including fixed-step Euler and RK4, the adaptive-step Dopri5 [50], and the Symplectic-Euler method [13]. The Symplectic-Euler method, intrinsically designed to be energy-conserving, is mathematically favorable for preserving the dynamic properties of Hamiltonian systems over long integration times. While observations indicate that solver selection does influence peak performance marginally, optimizing the solver was not the primary focus of this framework. For the sake of computational efficiency and consistency, we used the standard Euler solver across all main experiments.

3.1.3. Performance on Graph Classification

Results on human connectomes.

Table 7 provides the diagnostic performance across six disease-based cohorts, predicting the likelihood of neurological pathologies based on functional graph topologies. The empirical findings demonstrate that Nash-GNN exhibits decent accuracy in disease diagnosis, indicating its significant potential for clinical deployment in early computer-aided diagnosis.

TABLE 7.

Diagnosis accuracies (%) on disease-based datasets.

Dataset	GCN	GAT	GraphSAGE	GraphCON	GraphBel	GRAND	ACMP	HANG	Nash-GNN
ABIDE	54.85±0.56	54.29±0.26	54.41±0.56	54.51±0.31	54.72±0.51	62.79±12.79	54.85±0.64	51.92±0.97	62.78*±1.26
ADNI	76.62±3.76	72.36±3.41	75.10±3.4	71.24±2.05	91.75±8.64	78.10±16.51	92.68±7.68	91.47±4.17	95.82*±1.50
OASIS	85.95±0.05	86.06±0.22	86.01±0.23	86.23±0.27	85.93±0.25	85.54±3.10	85.79±0.23	85.89±0.35	89.22*±0.18
PPMI	68.02±11.57	64.96±7.33	68.02±11.14	64.54±9.31	71.86±4.05	71.43±1.85	72.45±2.21	70.91±2.24	74.03*±3.11

Discussion.

Atypical neural connectivity and neuron loss are established hallmarks of many neurological disorders [51], [52]. Moreover, the “prion-like” spreading mechanism, where pathological proteins propagate via structural and functional networks, has been widely documented [53], [54]. Consequently, the network topology plays a vital role in determining the dynamics trajectory of pathology propagation. Moving beyond the standard attention mechanisms in conventional GNNs [44], the continuous mean-field control framework in Nash-GNN allows us to capture the dynamic mechanisms of disease progression from a macroscopic system perspective, effectively simulating how pathological signals diffuse or react across distinct brain regions.

Results on TUDataset.

Table 8 presents the benchmark results on the widely used TUDataset. Nash-GNN demonstrates dominant performance across vastly different graph modalities, including molecules, bioinformatics, and social networks. These results verify that our adaptive reaction-diffusion control framework is universally applicable and highly scalable, outperforming models that are tailored to specific data structures.

TABLE 8.

Performace on TUDataset.

Dataset	MUTAG	NCI1	ENZYMES	D&D	PTC_FM	IMDB-B	PROTEINS
GCN	0.730±0.022	0.609±0.020	0.247±0.010	0.705±0.010	0.608±0.022	0.740±0.002	0.706±0.008
GAT	0.727±0.021	0.574±0.026	0.265±0.017	0.693±0.012	0.609±0.021	0.727±0.012	0.705±0.005
GraphSAGE	0.732±0.023	0.705±0.003	0.300±0.014	0.715±0.008	0.602±0.019	0.729±0.010	0.704±0.005
GCNII	0.728±0.022	0.691±0.003	0.444±0.022	0.706±0.008	0.616±0.013	0.694±0.007	0.695±0.010
AM-GCN	0.803±0.015	0.665±0.002	0.411±0.022	0.741±0.004	0.620±0.004	0.505±0.017	0.713±0.006
Nash-GNN	0.834*±0.019	0.749*±0.004	0.546*±0.015	0.770*±0.009	0.644*±0.003	0.751*±0.013	0.740*±0.005

3.2. Method Exploration: New Insights into Graph Learning Dynamics

In this section, we shift our focus to the learned diffusive mobility function ${\eta }_1$, as it governs the continuous message-passing mechanism (i.e., information transport) within our generalized RDM framework. For each graph dataset, Nash-GNN dynamically generates ${\eta }_1$ at the node level. Specifically, the macroscopic adaptive controller ${ℱ}_m$ utilizes an ICNN backbone to map the evolving mean-field density $\rho$ to the graph-specific mobility control patterns.

To intuitively interpret these continuous control patterns, we assume the underlying convex function can be approximated locally by a polynomial formulation. We compute the empirical polynomial power $\alpha$ at each graph node by applying univariate polynomial fitting to the relation ${\eta }_1(\rho )\propto {\rho }^{\alpha }$, and then average these degrees across the graph. Subsequently, we perform post-hoc analyses at both the macroscopic graph level and the microscopic node level.

Macroscopic analysis: Mobility vs. Homophily.

To further investigate how global information exchange dynamics correlate with intrinsic graph topology, we visualize the model’s performance dependency on the homophily ratio $h$ across various datasets, jointly with the ICNN depth $L$ of the adaptive controller ${ℱ}_m$ (Fig. 4, with $h$ on the horizontal axis and $L$ on the vertical axis). And the empirical polynomial power $\alpha$ effectively acts as a learned indicator of diffusive mobility.

Fig. 4.

Relationship between the homophily ratio $h$ (horizontal axis) and ICNN depth $L$ (vertical axis), colored by the empirical polynomial power $\alpha$. Larger and brighter bubbles correspond to higher performance outcomes across nine distinct datasets.

Results and Discussion:

As macroscopic analysis suggests, the effective strategy for graph learning is to promote unimpeded information diffusion on homophilic graphs while heavily dampening the diffusion between connected nodes with differing labels in heterophilic graphs. Fig. 4 illustrates that our model autonomously rediscovers this heuristic principle purely through variational optimization. For heterophilic datasets (e.g., Texas and Wisconsin), the model learns a remarkably low mobility $(\alpha \le 0.21)$. Within our MFC objective functional, this restricted mobility naturally penalizes excessive variation in density, effectively restraining transport in heterophilic settings. Finally, the model can safely tolerate a deeper ICNN adaptive controller (e.g., $L\ge 32$) to capture more nuanced mobility patterns without suffering from performance degradation (represented by the large, dark bubbles in the lower-left region). In contrast, highly homophilic datasets (e.g., Cora and Pubmed) exhibit high learned mobility $(\alpha \ge 0.74)$, promoting unimpeded information diffusion. In this highmobility regime, a larger $L$ quickly introduces excessive variance and triggers severe over-smoothing, leading to a sharp performance decline (represented by the small, light-colored bubbles in the lower-right region). Ultimately, this demonstrates that the learned mobility $\alpha$ autonomously regulates local information exchange and guides the adaptive selection of ICNN depth $L$, entirely without relying on hand-crafted attention weights.

Microscopic analysis: Node-Level mobility and neurodynamics.

Second, we extend this continuous dynamic analysis to individual graph nodes within fMRI-derived functional human connectomes. We hypothesize that the learned localized mobility of node embeddings, which serves as a computational proxy for a brain region’s functional dysregulation, aligns with the underlying disruption of large-scale brain networks observed during disease progression.

Results and Discussion:

We investigate the functional relevance of the learned node-wise mobility. In Fig. 5, the node size is rendered proportionally to the underlying mobility rate of the specific brain region. Remarkably, we observe that brain regions exhibiting high functional volatility, indicative of atypical information exchange dynamics, closely align with established clinical etiology. For Alzheimer’s disease, mounting empirical evidence demonstrates that disease progression is tightly coupled with early and severe functional disruption within the default mode network (DMN), a vulnerability largely driven by the accelerated spreading of underlying amyloid deposition [55]. Our data-driven MFC framework provides independent computational validation for this functional vulnerability: the nodes assigned the highest learned mobility predominantly cluster within the DMN, reflecting the network’s disproportionate functional dysregulation in response to AD pathology. Similarly, our continuous dynamic analysis reveals that (1) heightened functional volatility in the cerebellum is involved as a neurophysiological signature of Parkinson’s disease progression, and (2) aberrant information exchange dynamics within the dorsal attention network emerge as key functional markers for autism spectrum disorder. These promising findings indicate how explainable, continuous graph learning can elucidate complex functional disruptions, opening new windows into systems neuroscience and moving beyond the “black-box” nature of conventional GNNs.

Fig. 5.

The top ten significant regions associated with diseases.

4. Discussion and limitations

Discussion.

We explore the potential of alleviating oversmoothing issues in GNN, even with very deep network architecture. The experimental results are listed in Table 9. Unlike traditional GNNs, directly solving PDEs in our method enables fine-grained control over the speed and range of information propagation, allowing for more selective and localized information flow. This control helps prevent excessive information aggregation, ensuring that node representations do not neutralize too quickly, thus alleviating the over-smoothing problem. By doing so, this serves as evidence that Nash-GNN harmonizes individual node features with the macroscopic graph topology.

TABLE 9.

Performance across different numbers of layers.

# Layers	1	2	8	16	32	64	128
Texas	0.92591	0.90037	0.89037	0.91850	0.88333	0.90740	0.89185
Wisconsin	0.84000	0.85333	0.88667	0.88461	0.87333	0.88266	0.87615
Citeseer	0.76188	0.77313	0.74627	0.78507	0.75702	0.77611	0.78209

Limitations.

We list the average inference time on the Cora dataset (2708 nodes) for different models used in our study in Table 10. All graph PDE models employ the Euler solver, an integration time of 3, and a step size of 1. It is observed that our Nash-GNN uses a longer running time compared to non-PDE-based deep models (GCN, GAT, GraphSAGE, and GCNII). Regarding the comparison with counterpart PDE-based methods (e.g., GraphBel, ACMP, HANG), the computational overhead of Nash-GNN remains comparable, with all PDE-based models falling within a time range of roughly 10 ms to 35 ms per inference.

TABLE 10.

Model inference time (ms) comparison across various architectures.

Model	GCN	GAT	GraphSAGE	GraphCON	GraphBel	GRAND	ACMP	HANG	GCNII	Nash-GNN
Time(ms)	2.5570	4.5409	1.0505	2.2354	24.5252	10.228	27.1618	33.4651	3.0198	33.2494

5. Conclusions

In this paper, we introduced a unified variational framework grounded in mean-field control to fundamentally redefine the paradigm of graph representation learning. Moving beyond heuristic, discrete message-passing designs, we conceptualized the feature evolution on graphs as a continuous reaction-diffusion model governed by Hamiltonian mechanics. By parameterizing the underlying diffusive and reactive mobility patterns $\left({\eta }_1,{\eta }_2\right)$ via a macroscopic adaptive controller, our proposed Nash-GNN dynamically calibrates relational inductive biases that are specifically tailored to diverse graph topologies. Extensive empirical evaluations demonstrate that Nash-GNN not only establishes state-of-the-art performance across both homophilic and heterophilic benchmark datasets but also exhibits decent diagnostic accuracy on clinical human connectomes. More importantly, our continuous dynamic analyses bridge the gap between artificial graph learning mechanisms and biological pathology propagation, offering profound physical and medical insights into neurological disorders. We expect this physics-informed, adaptive control framework to pave a principled and mathematically rigorous new avenue for designing transparent, generalizable next-generation graph neural networks.

Biographies

Tingting Dan is currently an Assistant Professor with the Department of Psychiatry, The University of North Carolina, Chapel Hill. She received her PhD degree in computer science from South China University of Technology, Guangzhou, China, in 2023. Her research interests include machine learning, geometric deep learning, and medical image analysis.

Zhixuan Zhou is currently a PhD student with the Department of Computer Science, The University of North Carolina, Chapel Hill. His research interests include machine learning, geometric deep learning, medical image analysis, and system dynamics.

Won Hwa Kim is currently an Associate Professor in Graduate School of Artificial Intelligence (GSAI) / Computer Science and Engineering (CSE) / Medical Science and Engineering (MED) at Pohang University of Science and Technology (POSTECH). He received the BS degree in information and communication engineering from Sungkyunkwan University, in 2008, the MS degree in robotics, from KAIST, in 2010, and the PhD degree in computer sciences from the University of Wisconsin - Madison, in 2017. His research interests include interdisciplinary research that crosses cores in computer vision, machine learning and neuroscience.

Guorong Wu is currently an Associate Professor with the Department of Psychiatry and Computer Science, The University of North Carolina, Chapel Hill. He received the PhD degree in computer science and engineering from Shanghai Jiao Tong University, Shanghai, China. His research interests include machine learning, geometric deep learning, fast and robust analysis of large population data, computer-assisted diagnosis, and image-guided radiation therapy. In 2016, he was the recipient of the Independent Career Award from NIH for the contribution on noninvasive neurobiological basis for early diagnosis of Alzheimer’s disease.