SINDyG: Sparse Identification of Nonlinear Dynamical Systems from Graph-Structured Data, with Applications to Stuart-Landau Oscillator Networks

Technical Summary: SINDyG – Sparse Identification of Nonlinear Dynamical Systems from Graph-Structured Data

Problem Statement
The extraction of governing equations from data using sparse identification techniques, such as Sparse Identification of Nonlinear Dynamics (SINDy), has revolutionized modeling in fields ranging from neuroscience to climate science. However, standard SINDy approaches typically treat dynamical systems as monolithic entities, ignoring the underlying network topology that governs interactions between subsystems. When applied to graph-structured systems (e.g., neural networks, power grids, or social networks), standard SINDy treats all candidate interaction terms equally. This lack of structural priors often leads to the selection of "spurious terms"—interactions that do not reflect actual node-to-node connectivity—resulting in biased, less accurate, and overly complex models that fail to capture subtle emergent behaviors.

Methodology: SINDyG
To address these limitations, the authors propose SINDyG (Sparse Identification of Nonlinear Dynamical Systems from Graph-structured data). This method extends the standard SINDy framework by explicitly incorporating the network's adjacency structure into the sparse regression process.

The core innovation is a graph-informed regularization strategy. Instead of applying a uniform penalty to all candidate terms, SINDyG utilizes a penalty matrix $P$ derived from the network's adjacency matrix $A$. The process involves:

1. Connectivity Analysis: For each candidate term in the function library, the algorithm identifies source variables (nodes involved in the term) and calculates the mean connectivity ($m_{ck}$) between these sources and the sink node (the node whose dynamics are being modeled) using the adjacency matrix.
2. Penalty Formulation: A sigmoid-based penalty function $P[c,k]$ is constructed. Terms with low mean connectivity between source and sink variables receive a higher penalty (closer to 1), effectively discouraging the selection of interactions that violate the known network topology. Conversely, terms consistent with strong connections receive lower penalties.
3. Optimization: The method employs a modified Sequentially Thresholded Least Squares (STLSQ) algorithm. The objective function minimizes the reconstruction error of the derivatives while incorporating the graph-aware penalty:
   $$\min_{\Xi} \frac{1}{2} \| \dot{X} - \Theta(X)\Xi \|_2^2 + \lambda \| P \odot \Xi \|_2^2$$
   To solve this efficiently, the feature matrix and coefficients are scaled to transform the problem into a standard ridge regression framework, which is then solved iteratively with hard thresholding to enforce sparsity.

Key Contributions

• Graph-Aware Sparse Regression: The introduction of a modular penalty matrix that integrates prior knowledge of network connectivity directly into the coefficient estimation process.
• Improved Parsimony and Accuracy: By biasing the regression toward structurally plausible interactions, SINDyG reduces the selection of spurious terms, leading to simpler, more interpretable models.
• Modular Framework: The proposed graph-informed penalty is designed to be easily integrated with other symbolic regression solvers beyond SINDy.

Experimental Results
The authors validated SINDyG using Stuart-Landau (SL) oscillator networks, a model frequently used to describe the macroscopic oscillation of neuronal populations. Experiments were conducted on both simple three-node networks and complex general cases involving Erdős-Rényi (ER) and Scale-Free (SF) topologies with varying sizes and coupling strengths.

• Simple Case: In a three-node network, SINDyG achieved a model complexity ($\gamma$) of 32, matching the true model, whereas standard SINDy yielded a complexity of 62. SINDyG demonstrated superior coefficient accuracy (Mean Absolute Error of 0.0033 vs. 0.0155 for SINDy) and higher predictive $R^2$ scores on unseen trajectories.
• General Case: Sensitivity analyses revealed that as network size and coupling strength increased, standard SINDy performance degraded significantly, exhibiting higher complexity and error. In contrast, SINDyG maintained robustness, consistently producing models with lower complexity and higher generalization accuracy (higher test $R^2$, lower test MSE) across varying network sizes and edge weights.
• Computational Efficiency: While SINDyG involves additional steps to compute the penalty matrix, the overall training time remained comparable to standard SINDy, with the method demonstrating greater resilience to limited training data.

Significance and Claims
The paper claims that SINDyG represents a significant step forward in data-driven modeling of interconnected systems. By leveraging network topology to guide the discovery of governing equations, the method produces models that are not only more accurate but also more physically interpretable. The authors position SINDyG as a foundational step toward a broader class of "network-informed discovery algorithms." They suggest that integrating this graph-structured sparsity regularization into existing symbolic regression solvers (such as PySR) could enhance performance in noisy or real-world settings across diverse domains, including neuroscience, ecology, finance, and epidemiology. The authors modestly note that while the method was validated on Stuart-Landau models, future work is required to assess its efficacy on other nonlinear systems like Kuramoto or FitzHugh–Nagumo models.