Unveiling the Actual Performance of Neural-based Models for Equation Discovery on Graph Dynamical Systems

The ``black-box'' nature of deep learning models presents a significant barrier to their adoption for scientific discovery, where interpretability is paramount. This challenge is especially pronounced in discovering the governing equations of dynamical processes on networks or graphs, since even their topological structure further affects the processes' behavior. This paper provides a rigorous, comparative assessment of state-of-the-art symbolic regression techniques for this task. We evaluate established methods, including sparse regression and MLP-based architectures, and introduce a novel adaptation of Kolmogorov-Arnold Networks (KANs) for graphs, designed to exploit their inherent interpretability. Across a suite of synthetic and real-world dynamical systems, our results demonstrate that both MLP and KAN-based architectures can successfully identify the underlying symbolic equations, significantly surpassing existing baselines. Critically, we show that KANs achieve this performance with greater parsimony and transparency, as their learnable activation functions provide a clearer mapping to the true physical dynamics. This study offers a practical guide for researchers, clarifying the trade-offs between model expressivity and interpretability, and establishes the viability of neural-based architectures for robust scientific discovery on complex systems.

Logic Explanation of AI Classifiers by Categorical Explaining Functors

The most common methods in explainable artificial intelligence are post-hoc techniques which identify the most relevant features used by pretrained opaque models. Some of the most advanced post hoc methods can generate explanations that account for the mutual interactions of input features in the form of logic rules. However, these methods frequently fail to guarantee the consistency of the extracted explanations with the model's underlying reasoning. To bridge this gap, we propose a theoretically grounded approach to ensure coherence and fidelity of the extracted explanations, moving beyond the limitations of current heuristic-based approaches. To this end, drawing from category theory, we introduce an explaining functor which structurally preserves logical entailment between the explanation and the opaque model's reasoning. As a proof of concept, we validate the proposed theoretical constructions on a synthetic benchmark verifying how the proposed approach significantly mitigates the generation of contradictory or unfaithful explanations.

Exact Computation of Any-Order Shapley Interactions for Graph Neural Networks

Albeit the ubiquitous use of Graph Neural Networks (GNNs) in machine learning (ML) prediction tasks involving graph-structured data, their interpretability remains challenging. In explainable artificial intelligence (XAI), the Shapley Value (SV) is the predominant method to quantify contributions of individual features to a ML model's output. Addressing the limitations of SVs in complex prediction models, Shapley Interactions (SIs) extend the SV to groups of features. In this work, we explain single graph predictions of GNNs with SIs that quantify node contributions and interactions among multiple nodes. By exploiting the GNN architecture, we show that the structure of interactions in node embeddings are preserved for graph prediction. As a result, the exponential complexity of SIs depends only on the receptive fields, i.e. the message-passing ranges determined by the connectivity of the graph and the number of convolutional layers. Based on our theoretical results, we introduce GraphSHAP-IQ, an efficient approach to compute any-order SIs exactly. GraphSHAP-IQ is applicable to popular message passing techniques in conjunction with a linear global pooling and output layer. We showcase that GraphSHAP-IQ substantially reduces the exponential complexity of computing exact SIs on multiple benchmark datasets. Beyond exact computation, we evaluate GraphSHAP-IQ's approximation of SIs on popular GNN architectures and compare with existing baselines. Lastly, we visualize SIs of real-world water distribution networks and molecule structures using a SI-Graph.