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(logN). 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.
Table 1: Relative L2 errors on two Helmholtz settings. The manufactured-solution block follows the protocol of prior work: Ω=[−1,1]3 , u=sin(4πx1)sin(4πx2)sin(3πx3) , with the corresponding source term obtained by substituting u into the Helmholtz equation, and the number of collocation points Nc=323 . The dipole block is a 2.4 GHz radiation problem with a singular excitation and absorbing truncation, with both methods run under identical settings. First-block baselines are as reported in Duan et al. (2025) ; all remaining values are from our own runs using the authors’ recommended settings.
Figure 1: Overview of PE-EK-PINN structure.
Scenario
Base config.
PE-EK-PINN construction
Transformation
Dipole array
Single dipole
1→2×2→4×4→8×8→16×16
Translation
Composite line geometry
2λ line
2λ line → {Cross, 5-point star}
Translation + rotation
Cross array (Section B.2 )
0.5λ line
0.5λ line → Cross →2×2→4×4
Translation + rotation
Table 2: Overview of the evaluated source configurations.
Method
Ndipoles
Nkernels
Stage T.
Cum. T.
MSE
Rel. L2
PE-PINN
1
2
00:19:47
N/A
9.48×10−6
2.14×10−2
PE-EK-PINN
2×2
4
00:26:51
00:46:38
2.49×10−5
1.55×10−2
PE-EK-PINN
4×4
4
00:26:53
01:13:31
1.86×10−4
1.62×10−2
PE-EK-PINN
8×8
4
00:27:29
01:41:00
3.78×10−3
2.71×10−2
PE-EK-PINN
16×16
4
00:29:35
02:10:35
3.20×10−1
9.43×10−2
PE-PINN
2×2
8
01:07:48
N/A
5.83×10−5
2.37×10−2
Table 3: Accuracy and training time for the dipole-array experiments. PE-PINN results for the 2×2 and 4×4 configurations serve as an ablation baseline. Cumulative time includes all preceding cascade stages starting from the single-dipole model.
Figure 2: Visualized results of the 16×16 dipole-array and 5-point star finite-line experiments. Please refer to Appendix B.3 for the complete visualized results.
Method
Config.
Kernels
Stage T.
Cum. T.
MSE
Rel. L2
PE-PINN
Line
10
01:37:08
N/A
8.85×10−5
4.12×10−2
PE-EK-PINN
Two-line cross
2
00:16:29
01:53:37
1.18×10−4
3.34×10−2
PE-PINN
Two-line cross
20
03:13:19
N/A
1.87×10−3
1.33×10−1
PE-PINN
Line wide
10
02:21:57
N/A
2.07×10−4
7.56×10−2
PE-EK-PINN
5-point star
5
00:39:18
03:01:14
1.24×10−3
6.32×10−2
Table 4: Accuracy and training time for the finite-line-source experiments. The wider-domain line kernel is trained on Ω=[−3.6,3.6]2 to cover the coordinate range reached after rotation.
Appendix figures & tables6 assets
Supplementary material from the paper’s appendix.
Appendix
Method
Config.
Kernels
Stage T.
Cum. T.
MSE
Rel. L2
PE-PINN
0.5λ line
4
00:42:59
00:42:59
6.38×10−5
6.39×10−2
PE-EK-PINN
Cross
2
00:16:26
00:59:25
1.07×10−4
4.18×10−2
PE-PINN
Cross
8
01:00:50
N/A
5.30×10−4
9.33×10−2
PE-EK-PINN
2×2
4
00:31:44
01:31:09
8.00×10−4
5.07×10−2
PE-PINN
2×2
32
04:05:07
N/A
3.20×10−3
1.01×10−1
PE-EK-PINN
4×4
4
00:32:24
02:03:33
7.57×10−3
5.90×10−2
Appendix
Table 5: Accuracy and training time for the hierarchical cross-array experiments. PE-PINN baselines use primitive point-source kernels only.
Figure 3: Visualized results of the dipole-array experiments.
Figure 4: Training-time scaling for direct PE-PINN and PE-EK-PINN. Stage time denotes the training cost of the current cascade level, while cumulative time includes all preceding cascade stages. Direct PE-PINN times for the 8×8 and 16×16 arrays are extrapolated based on the observed near-linear scaling with active-kernel count.
Figure 5: Visualized results of the composite line-source experiments.
Figure 6: Visualized results the cross-array experiments.
Figure 7: Training-time scaling for the hierarchical cross-array experiment. Stage time denotes the training cost of the current cascade level, while cumulative time includes all preceding stages from the elementary 0.5λ line-source model. The direct PE-PINN time for the 4×4 cross array is extrapolated based on the observed near-linear scaling with active kernel count.
Division of Decision and Control Systems, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, Stockholm, 100 44, Sweden