arXiv subjects

Jonas Naujoks

Publications and source records attributed to Jonas Naujoks.

3 recordsLinked to original sources

arXiv2026

Value or Attention: Investigating Layer-wise Relevance Propagation for Graph Attention Networks

Graph Attention Networks (GATs) are increasingly used in domains where scientists need to read a model's reasoning, from medicine to materials science, yet the routing they learn remains opaque to most explainers. Layer-wise Relevance Propagation (LRP) follows the model's computation, but in an attention layer the relevance reaching a node is the sum of two mechanically different quantities: its content relevance through the value path and its routing relevance through the attention path. Existing rules either drop one or add them into a single number. We write a single product rule with free weights on the two factors that recovers current approaches as special cases. The value path measures what content a node supplied against a zero baseline; the attention path measures how the node shaped what was passed relative to its neighborhood, and its sign records whether the node's features moved the gating the way the prediction demanded. On a ground-truth classification task, we show that the same node receives negative, positive and near-zero relevance under three settings of the rule. The separated relevance maps yield a hypothesis about the model: it gains by not reading that node. We give reading rules for both paths and argue that they should be interpreted separately.

cs.LG↗

arXiv2026

LieSolver: PDE-Constrained Learning for IBVPs via Lie Symmetries

Initial-boundary value problems (IBVPs) provide the essential framework for modelling a wide range of phenomena in physics and engineering. We introduce a novel method for efficiently solving IBVPs using Lie symmetries to enforce the associated partial differential equation (PDE) exactly by construction. By leveraging symmetry transformations, our model embeds the underlying physical laws and learns the solution solely from initial and boundary data. Consequently, the boundary loss directly quantifies domain-wide error, enabling rigorous error estimation for well-posed IBVPs. We implement LieSolver and demonstrate its application to linear homogeneous PDEs, showing that it outperforms physics-informed neural networks (PINNs) in both speed and accuracy while yielding compact models. Overall, our approach significantly enhances the efficiency and reliability of predictions for PDE-constrained problems.

cs.LG↗

arXiv2026

Building Trust in PINNs: Error Estimation through Finite Difference Methods

Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their flexibility, PINNs offer limited insight into how their predictions deviate from the true solution, hindering trust in their prediction quality. We propose a lightweight post-hoc method that addresses this gap by producing pointwise error estimates for PINN predictions, which offer a natural form of explanation for such models, identifying not just whether a prediction is wrong, but where and by how much. For linear partial differential equations, the error between a PINN approximation and the true solution satisfies the same differential operator as the original problem, but driven by the PINN's PDE residual as its source term. We solve this error equation numerically using finite difference methods requiring no knowledge of the true solution. Evaluated on several benchmark PDEs, our method yields accurate error maps at low computational cost, enabling targeted and interpretable validation of PINNs.

cs.LG↗