PIEFS: Physics-Informed Eigenfunction Features with Learnable Scaling

• URL: http://arxiv.org/abs/2607.03692v2
• Date: Tue, 07 Jul 2026 08:23:30 GMT
• Title: PIEFS: Physics-Informed Eigenfunction Features with Learnable Scaling
• Abstract summary: We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS)<n>PIEFS is a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy.
• Score: 42.058442499492536
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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