{"ID":5937881,"CreatedAt":"2026-07-07T03:14:33.014478982Z","UpdatedAt":"2026-07-07T03:46:50.908361658Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.03692","arxiv_id":"2607.03692","title":"PIEFS: Physics-Informed Eigenfunction Features with Learnable Scaling","abstract":"Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy. In PIEFS, scalar coordinate maps are trained under empirical Gram orthogonality, a supervised linear readout, and a Dirichlet penalty in which the input gradient is transformed by a learnable metric $A(x)=Λ(x)U(x)$. The diagonal factor $Λ(x)$ controls anisotropic scaling, while the orthogonal factor $U(x)$ is parameterized by a structured product of Givens rotations. This construction yields task-adaptive Dirichlet-regularized coordinates rather than eigenfunctions of a fixed supervision-independent operator. Experiments on synthetic, tabular, and image-based benchmarks study the effect of identity, diagonal, and rotation-scaling metrics, and compare the resulting coordinates with classical baselines and NeuralEF. The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.","short_abstract":"Spectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework...","url_abs":"https://arxiv.org/abs/2607.03692","url_pdf":"https://arxiv.org/pdf/2607.03692v1","authors":"[\"Varvara Nazarenkko\",\"Timur Lidzhiev\",\"Alexander Tarakanov\"]","published":"2026-07-04T03:55:40Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"math.NA\",\"math.ST\"]","methods":"[]","has_code":false}