# PE-EK-PINN: Physics Embedding with Evolving Kernel for Scalable Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2609.38023 · HTML: https://mlsanity.com/paper/2609.38023/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 20 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 12 of 12)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [RoPINN: Region Optimized Physics-Informed Neural Networks](https://mlsanity.com/paper/2405.14369/) (2024)
- [Physics-Informed Neural Networks: Minimizing Residual Loss with Wide Networks and Effective Activations](https://mlsanity.com/paper/2405.01680/) (2024)

## Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) embed governing equations into deep learning, but enforce them only through loss residuals, leaving highly oscillatory wave behavior to be discovered by optimization. As a result, methods that achieve relative L2 errors below 10-3 on standard manufactured Helmholtz benchmarks can fail on practical radiation problems involving singular excitations, absorbing boundaries, and wave fields spanning tens of wavelengths. Architectural physics embedding addresses this limitation by factorizing the field into analytically derived oscillatory kernels and learnable envelopes. However, the kernel dictionary must be manually constructed and scales with the number of elementary units, growing exponentially with the depth of hierarchically structured systems such as antenna arrays and metasurfaces. We propose PE-EK-PINN (Physics Embedded with Evolving Kernels), which treats physics kernels as reusable learned representations rather than fixed analytical inputs. A converged subsystem field is frozen and promoted to an evolved kernel, whose transformed copies are reused to represent higher-level configurations without deriving new governing equations. The resulting hierarchy makes the peak number of active kernels independent of system size and reduces cumulative training cost from O(N) to O(log N). Experiments on dipole arrays, composite line-source geometries, and cross arrays demonstrate the dramatic training cost reduction, while achieving a reduced or comparable relative L2 error. One notable example is PE-EK-PINN solves a 256-dipole array more than 30 times faster than direct PE-PINN.