arXiv:2605.28578v1 Announce Type: new 
Abstract: We propose the Tikhonov layer, a graph neural network layer that is interpretable by design: once trained, its learned parameters directly reveal which node features and which aspects of the graph topology were leveraged for prediction. In practice, the layer's propagation matrix takes the closed-form $R = (p(L)+Q)^{-1} Q$, where $L$ is the normalized graph Laplacian, $Q = diag(q_1,...,q_n)$ a learnable diagonal matrix of positive node-importance scores, and $p(\cdot)$ a learnable polynomial. For any input feature $x$, the layer output $Rx$ is the exact minimizer of a generalized graph Tikhonov problem that trades off node-level data fidelity against a topology-driven regularization penalty. The learned pair $\{\{q_i\},p\}$ constitutes a built-in explanation: large $q_i$ indicates that node $i$'s own features drive the prediction, while small $q_i$ signals reliance on the local graph topology; the shape of $p$ reveals whether homophily, heterophily, or a band-pass response is exploited. Expressivity is preserved by routing complexity through a dedicated, arbitrarily deep Q-network that produces the importance scores, while the Tikhonov layer itself remains transparent. We prove that distinct node-importance matrices yield distinct propagation operators, structurally coupling the explanation to the computation. Additionally, the Tikhonov layer provides, in a single layer, a global receptive field, mitigating both oversmoothing and oversquashing. Experiments on standard graph classification benchmarks confirm that the model matches (and sometimes outperforms) opaque baselines while producing interpretable and faithful explanations.

قدمت دراسة جديدة طبقة تيخونوف، وهي طبقة شبكة عصبية رسومية مصممة لتعزيز القابلية للتفسير من خلال الكشف عن تأثير ميزات العقد وتوبولوجيا الرسم البياني على التنبؤات. تستخدم هذه الطبقة مصفوفة انتشار مستمدة من لاباسيان الرسم البياني المنظم ومصفوفة قطرية قابلة للتعلم من درجات أهمية العقد، مما يسمح بتنبؤات دقيقة مع توفير تفسيرات مدمجة لقرارات النموذج.

Un nuevo estudio ha presentado la capa Tikhonov, una capa de red neuronal gráfica diseñada para mejorar la interpretabilidad al revelar la influencia de las características de los nodos y la topología del grafo en las predicciones. Esta capa emplea una matriz de propagación derivada del laplaciano normalizado del grafo y una matriz diagonal aprendible de puntajes de importancia de nodos, permitiendo predicciones precisas mientras proporciona explicaciones integradas para las decisiones del modelo.

Une nouvelle étude a introduit la couche Tikhonov, une couche de réseau de neurones graphique conçue pour améliorer l'interprétabilité en révélant l'influence des caractéristiques des nœuds et de la topologie du graphe sur les prédictions. Cette couche utilise une matrice de propagation dérivée du laplacien normalisé du graphe et une matrice diagonale apprenante des scores d'importance des nœuds, permettant des prédictions précises tout en fournissant des explications intégrées pour les décisions du modèle.

A new study has introduced the Tikhonov layer, a graph neural network layer designed to enhance interpretability by revealing the influence of node features and graph topology on predictions. This layer employs a propagation matrix derived from the normalized graph Laplacian and a learnable diagonal matrix of node-importance scores, enabling precise predictions while providing built-in explanations for the model's decisions.

A Generalized Tikhonov Layer for Interpretable-by-design Graph Neural Networks

arXiv:2606.11870v1 Announce Type: cross 
Abstract: Machine learning is increasingly applied to accelerate the discovery of novel materials by exploring large compositional and structural design spaces. Yet, the scarcity of high-quality data and the frequent need for out-of-distribution prediction introduce substantial uncertainty, making the assessment of model reliability essential. In this work, we investigate uncertainty quantification as a means to evaluate model confidence in the context of permanent magnet research. In a first study, we benchmark classical and modern machine learning models for predicting intrinsic magnetic properties, focusing on the quality of their uncertainty estimates. We apply Gaussian negative log-likelihood loss and dropout-based Bayesian approximation as practical strategies for estimating predictive uncertainty. In a second study, we transfer these architectural features for uncertainty estimation to a more complex task: predicting coercivity from microstructural information using a graph neural network. Together, these studies demonstrate that uncertainty quantification not only enhances the trustworthiness of predictions but is also transferable across different modeling tasks.

دراسة حديثة نُشرت على arXiv تستكشف تطبيق الشبكات العصبية الواعية بالشكوك في نمذجة الخصائص المغناطيسية للمواد، مع التركيز بشكل خاص على المغناطيسات الدائمة. تبحث الدراسة في تقييم قدرات النماذج الكلاسيكية والحديثة في التعلم الآلي وقدرتها على التنبؤ، باستخدام تقنيات مثل خسارة اللوغاريتم السلبية الغاوسية والتقريب البايزي القائم على التسرب.

Un estudio reciente publicado en arXiv investiga la aplicación de redes neuronales conscientes de la incertidumbre en la modelización de las propiedades magnéticas de los materiales, centrándose especialmente en los imanes permanentes. La investigación evalúa las capacidades predictivas y las estimaciones de incertidumbre de modelos de aprendizaje automático clásicos y modernos, utilizando técnicas como la pérdida de log-verosimilitud negativa gaussiana y la aproximación bayesiana basada en el dropout.

Une étude récente publiée sur arXiv examine l'application de réseaux de neurones conscients de l'incertitude dans la modélisation des propriétés magnétiques des matériaux, en se concentrant particulièrement sur les aimants permanents. La recherche évalue les capacités prédictives et les estimations d'incertitude des modèles d'apprentissage automatique classiques et modernes, en utilisant des techniques telles que la perte de log-vraisemblance négative gaussienne et l'approximation bayésienne basée sur le dropout.

A recent study published on arXiv investigates the application of uncertainty-aware neural networks in modeling the magnetic properties of materials, particularly focusing on permanent magnets. The research benchmarks classical and modern machine learning models to assess their predictive capabilities and uncertainty estimates, utilizing techniques such as Gaussian negative log-likelihood loss and dropout-based Bayesian approximation.

A Generalized Tikhonov Layer for Interpretable-by-design Graph Neural Networks

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