# PIEFS: Physics-Informed Eigenfunction Features with Learnable Scaling

arXiv: https://arxiv.org/abs/2607.03692 · HTML: https://mlsanity.com/paper/2607.03692/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 16 references that are in the map.
- First posted on arXiv 4 Jul 2026.

## Builds on (most cited first, 2 of 2)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Unsupervised Neural Networks for Quantum Eigenvalue Problems](https://mlsanity.com/paper/2010.05075/) (2020)

## Abstract (arXiv)

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)=Lambda(x)U(x). The diagonal factor Lambda(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.