PaperlayerView exact revision on arXiv

Source-linked AI summary

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

Huiwen Zhang, Feng Ye, Chu Ma

arXiv:2609.38023v1cs.AI

TL;DR

Practical radiation fields can challenge PINNs that perform well on manufactured Helmholtz problems, and manually designed physics kernels limit architectural physics embedding at scale. PE-EK-PINN turns learned subsystem fields into reusable kernels for hierarchical configurations. Across the evaluated structured systems, it validates reduced training growth while improving training efficiency and reconstruction accuracy over direct PE-PINN.

  • Problem

    Manufactured Helmholtz performance does not necessarily transfer to practical radiation problems, while direct physics-embedded kernel sets grow with system size.

    Source: §1 · p. 1, §3.2 · p. 4

  • Method

    PE-EK-PINN freezes a converged subsystem field as an evolved kernel and reuses transformed copies to represent higher-level configurations.

    Source: §1 · p. 2

  • Results

    Experiments on dipole arrays, cross arrays, and composite line geometries validate bounded active-kernel counts and reduced cumulative training growth, with improved efficiency and reconstruction accuracy over direct PE-PINN.

    Source: §6 · p. 9

  • Takeaways & Limitations

    Reusable learned kernels show potential for scaling physics-informed wave modeling to larger structured systems.

    Source: §6 · p. 9

    Evolved-kernel errors propagate upward and become more noticeable at greater hierarchy depths, while configurations without repeated substructure offer limited compression.

    Source: §5.3 · p. 9

Abstract

from arXiv · show

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 $L_2$ 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 $L_2$ error. One notable example is PE-EK-PINN solves a $256$-dipole array more than 30 times faster than direct PE-PINN.

1 INTRODUCTION

Practical radiation problems expose a gap between manufactured Helmholtz benchmarks and real wave fields, while manually designed physics kernels limit architectural physics embedding at scale. PE-EK-PINN reuses learned subsystem fields as kernels, reducing training cost for large structured systems.

1 INTRODUCTION

  • CoPINN’s relative L2 error rises from 5.0 × 10^-3 on the manufactured benchmark to 9.95 × 10^-1 on a 2.4 GHz dipole problem.The practical problem includes a singular source, absorbing boundaries, and a domain spanning many wavelengths.Source: §1 · p. 1
  • PINNs rely on optimization to discover oscillatory solutions, and practical radiation fields can include singular sources, absorbing boundaries, and many wavelengths.Manufactured Helmholtz benchmarks can therefore overstate performance on practical radiation problems.Source: §1 · p. 1
  • PE-PINN embeds analytically derived oscillatory kernels and learnable envelopes, reducing spectral bias on the dipole problem to a relative L2 error of 2.14 × 10^-2.Its manually constructed kernel dictionary grows with source-configuration size, creating a scalability bottleneck.Source: §1 · pp. 1–2
  • PE-EK-PINN freezes a converged subsystem field as an evolved kernel and reuses transformed copies to build higher-level configurations.In homogeneous media, rigid-motion invariance of the Helmholtz operator preserves physical consistency during reuse.Source: §1 · p. 2
  • A 256-dipole array trains in 2 h 11 m versus an extrapolated 71 h for direct PE-PINN, with a fourfold reduction in relative L2 error.The paper evaluates dipole arrays, composite line-source geometries, and cross arrays.Source: §1 · p. 2
  • Neural operators can scale repeated PDE evaluations but require high-fidelity training data and face challenges with oscillatory fields and singularities.The paper identifies future work on using its solutions as training data and extending kernel factorization to neural operators.Source: §1 · p. 2

2 RELATED WORK

Prior PINN remedies improve optimization, computational cost, or generic function approximation, but do not embed wave physics itself. Manufactured-solution benchmarks also differ substantially from practical radiation settings.

2 RELATED WORK

  • Existing PINN remedies target cost, domain size, optimization conditioning, or representation, while leaving wave structure to be learned from coordinates.Examples include separable representations, domain decomposition, adaptive weighting and sampling, Fourier features, and periodic activations.Source: §2 · pp. 2–3
  • Manufactured Helmholtz benchmarks use smooth, separable fields on simple domains, unlike radiation problems with singular sources, absorbing boundaries, and many wavelengths.Performance on manufactured benchmarks therefore does not necessarily transfer to practical radiation settings.Source: §2 · p. 3
  • PE-PINN embeds analytically derived oscillatory kernels and learnable envelopes, but its manually constructed kernel set grows with source-configuration size.Its representation also uses incident/scattered-field separation and material-aware domain decomposition.Source: §2 · p. 3

3 PRELIMINARIES AND SCALABILITY CHALLENGE

The paper formulates wave fields from hierarchically repeated sources and explains how direct PE-PINN representations scale with the number of elementary units. Reusing subsystem solutions is motivated as a way to exploit that hierarchy.

3.1 PRELIMINARIES

  • The setup considers a two-dimensional complex electric field satisfying the homogeneous Helmholtz equation, with excitation imposed at source-associated locations.The wavenumber is k = 2π/λ, and a first-order radial absorbing condition emulates an unbounded medium.Source: §3.1 · pp. 3–4
  • PE-PINN represents the field with analytic propagation kernels, learnable smooth envelopes, and spatial gates that limit each component to its validity region.For a point-like source, the primitive kernel has a spherical-wave form; the field’s rapid phase is encoded in the kernel.Source: §3.1 · p. 4
  • Structured source systems are built recursively from translated copies of smaller subsystems, so elementary-unit counts grow geometrically with hierarchy depth.The formulation also allows general rigid transformations, although the discussion focuses on translations.Source: §3.1 · p. 4

3.2 THE KERNEL SCALABILITY BOTTLENECK

  • Direct PE-PINN cost is linear in the number of elementary units and exponential in hierarchy depth, with larger kernel sets also making optimization more difficult.The cost estimate assumes roughly constant training epochs as the hierarchy expands.Source: §3.2 · p. 4
  • A learned lower-level subsystem field can be reused as one component at higher levels, targeting O(ℓ) trainable branches instead of O(Iℓ).This uses the known spatial relationship among translated sub-configurations while retaining architectural physics embedding.Source: §3.2 · pp. 4–5

4 PE-EK-PINN

PE-EK-PINN builds a hierarchy by freezing converged subsystem fields as reusable physics kernels, then reusing transformed copies at higher levels. This keeps peak active-kernel counts independent of system size and reduces cumulative training-cost dependence from O(N) to O(log N).

4.1 OVERVIEW

  • At each hierarchy level, PE-EK-PINN freezes the converged field representation as a reusable kernel for learning the next-level configuration.Previously learned representations are preserved and reused rather than relearned.Source: §4.1 · p. 5

4.2 EVOLVING PHYSICS KERNELS

  • A converged subsystem field becomes a learned, composite kernel that captures dominant oscillations, leaving the next-level network to learn a comparatively smooth envelope.The optimized parameters are frozen before the learned field is promoted.Source: §4.2 · p. 5

4.3 CASCADED KERNEL REUSE

  • Translated evolved kernels represent higher-level configurations, while learned envelopes capture subsystem interactions and satisfy source and outer-boundary conditions.Reuse across locations relies on Helmholtz translation invariance in a homogeneous medium with constant wavenumber.Source: §4.3 · pp. 5–6

4.4 TRAINING AND COMPUTATIONAL EFFICIENCY

  • Peak active kernels are independent of system size, while the hierarchy reduces cumulative training-cost dependence from O(N) to O(log N).The cumulative-cost scaling assumes approximately constant epochs per level and per-iteration cost linear in active kernels.Source: §4.4 · p. 6

5 EXPERIMENTS AND EVALUATION

The experiments evaluate PE-EK-PINN on dipole arrays and composite finite-line geometries, then discuss its scaling behavior and accuracy constraints. Across these tests, kernel reuse keeps active-kernel counts bounded, while accuracy and training cost vary with configuration and hierarchy depth.

5.1 EXPERIMENT SETUP

  • Experiments use analytical source fields, exclude singular source regions from PDE collocation, and compare complex relative L2 error with MSE as a secondary metric.The evaluations use free-space configurations at 2.4 GHz, a shared PE-PINN backbone, and fixed training hyperparameters.Source: §5.1 · p. 7, §5.2 · p. 7

5.2.1 DIPOLE ARRAY

  • 1.55 × 10−2, 1.62×10−2, and 2.71×10−2 relative L2 errors through 8×8 rise to 9.43×10−2 at 16×16.The dipole-array results are reported in Table 3.Source: §5.2.1 · p. 7, Table 3 · p. 8
  • 26.9 to 29.6 minutes of PE-EK-PINN stage time span 2×2 to 16×16 arrays despite a 64× increase in physical dipoles.PE-PINN training time increases 3.98× when active kernels rise from 8 to 32.Source: §5.2.1 · p. 8

5.2.2 COMPOSITE FINITE-LINE GEOMETRIES

  • 6.32×10−2 relative L2 error is achieved for the 5-point star with 0:39:18 of stage training; no PE-PINN baseline is reported.The wider-domain base line kernel increases training time by 1.46× and its base relative L2 error rises from 4.12 × 10−2 to 7.56×10−2.Source: Table 4 · p. 9, §5.2.2 · p. 9

5.3 DISCUSSION

  • A converged subsystem field becomes a reusable kernel, replacing multiple trainable envelope branches with a fixed component; speedup improves with hierarchy depth.The observed speedup grows more slowly than the reduction in kernel count.Source: §5.3 · p. 9
  • Approximation error in frozen kernels propagates upward, so base-kernel accuracy matters more as hierarchy depth increases.Later envelopes can compensate for interactions among reused components but cannot correct inaccuracies internal to a frozen kernel.Source: §5.3 · p. 9
  • Kernel reuse assumes translation invariance in homogeneous media and reliable evaluation within the MLP kernel’s training coordinate range.The method also assumes a known hierarchy, limiting compression for configurations without repeated substructure.Source: §5.3 · p. 9

6 CONCLUSION

PE-EK-PINN promotes learned subsystem fields into reusable physics kernels, keeping active-kernel counts bounded and cumulative training growth logarithmic. Experiments on dipole arrays, cross arrays, and composite line geometries validate the scaling and report improved training efficiency and reconstruction accuracy over direct PE-PINN.

  • PE-EK-PINN promotes learned subsystem fields into reusable physics kernels, keeping active-kernel counts bounded and cumulative training growth at O(log N).The conclusion contrasts this growth with O(N).Source: §6 · p. 9
  • Experiments on dipole arrays, cross arrays, and composite line geometries validate the scaling and report improved training efficiency and reconstruction accuracy over direct PE-PINN.Source: §6 · p. 9
  • The results demonstrate the potential of reusable learned kernels for scaling physics-informed wave modeling to larger structured systems.Source: §6 · p. 9

A SOURCE DEFINITIONS AND ANALYTICAL REFERENCE SOLUTIONS

The appendix specifies analytical references, normalization conventions, and boundary constraints for dipole and uniform finite-line sources. It describes singularity handling and the kernel assignments used for finite-line representations.

A SOURCE DEFINITIONS AND ANALYTICAL REFERENCE SOLUTIONS

  • The appendix defines closed-form source fields and conventions used to prescribe source constraints and evaluate predictions against analytical references.It states that λ is wavelength and k = 2π/λ, and identifies H(2) as the Hankel function of the second kind with outgoing-wave behavior under the stated time convention.Source: A SOURCE DEFINITIONS AND ANALYTICAL… · p. 11

A.1 DIPOLE SOURCE

  • Dipole fields are prescribed on a circular source boundary while the Helmholtz residual is enforced outside a larger exclusion region to avoid the source singularity.Dipole-array reference fields are formed by coherent superposition of normalized element fields with identical unit excitations.Source: §A.1 · pp. 11–12

A.2 UNIFORM FINITE LINE SOURCE

  • Finite-line fields are integrated along the segment, prescribed on a capsule boundary, and normalized by RMS magnitude over sampled source points.The source constraint also uses the outward normal derivative; 2λ and 0.5λ lines use ten and four spherical kernels, respectively, as representation components rather than physical sources.Source: §A.2 · p. 12

A.3 TRAINING CONSTRAINTS

The experiments enforce source agreement and the Helmholtz equation while using a radial outgoing condition on the outer boundaries to better match dominant wave propagation.

A.3 TRAINING CONSTRAINTS

  • The source loss enforces agreement with the analytical field on each source boundary.For dipoles, field matching alone defines the source constraint.Source: §A.3 · p. 13
  • For finite line sources, the source constraint also enforces the outward normal derivative because the boundary lies closer to the singular support.Source: §A.3 · p. 13
  • The Helmholtz residual is enforced at collocation points outside the exclusion regions.Source: §A.3 · p. 13
  • A first-order radial outgoing condition is imposed on outer boundaries because waves reach them mainly radially, making normal-derivative absorption less accurate near corners.The radial formulation follows rays from the configuration center and reduces artificial reflections.Source: §A.3 · p. 13

B EVALUATION RESULTS

The evaluation section describes PINN baselines that modify residual constraints, network factorization, optimization regions, or training strategies.

B.1 BASELINES

  • PINN embeds PDE residuals and boundary and initial conditions in neural-network optimization, learning continuous fields from collocation points without labeled solutions or an explicit mesh.Source: §B.1 · p. 13
  • gPINN adds input derivatives of the PDE residual as differential constraints intended to improve solution accuracy.Source: §B.1 · p. 13
  • SPINN factorizes inputs across coordinate axes and combines one-dimensional subnetworks; SPINN (m) keeps this separable form with a modified-MLP backbone.Source: §B.1 · p. 13
  • AHD-PINN analyzes residual-loss minimization and activation criteria, while RoPINN extends point-wise optimization to continuous neighborhood regions.AHD-PINN's results apply under stated conditions; RoPINN targets generalization, particularly for hidden higher-order constraints.Source: §B.1 · p. 13
  • FPINN uses fuzzy neural-network components for robustness to ambiguous or inaccurate data, while CoPINN schedules optimization from easier to harder regions using residual-gradient difficulty estimates.Source: §B.1 · p. 14

B.2 CROSS ARRAY

The cross-array experiments show that hierarchical kernel reuse reduces training time and improves accuracy over PE-PINN as configurations grow.

B.2 CROSS ARRAY

  • The hierarchy reuses a 0.5λ line-source kernel to form a cross, then a 2 × 2 array, then a 4 × 4 array, with rotation and translation.Source: §B.2 · p. 14
  • 7.72× speedup at the 2 × 2 stage (2.69× end-to-end) accompanies a relative L2 error reduction from 1.01 × 10−1 to 5.07 × 10−2; a single cross is 3.70× faster with error reduced from 9.33×10−2 to 4.18×10−2.Stage time remains approximately 32 minutes at both 2 × 2 and 4 × 4 despite a fourfold increase in physical crosses.Source: Table 5 · p. 14, §B.2 · p. 14
  • The hierarchy reaches only 16 physical elements; relative L2 error rises from 4.18 × 10−2 to 5.90 × 10−2 across levels, while the base stage uses 43 of 124 cumulative minutes.The first cascade level improves on its base model's 6.39 × 10−2 error.Source: §B.2 · p. 14

B.3 VISUALIZATION RESULTS

Visualizations show close qualitative agreement between analytical and PE-EK-PINN fields across dipole, line-source, and cross-array configurations, while training-time comparisons show much slower cost growth than direct PE-PINN.

B.3 VISUALIZATION RESULTS

  • A pretrained 2λ line-source representation is reused by translation and rotation to form two-line and pentagram geometries, whose analytical and learned fields show similar real and imaginary wave structures.Complex-error maps visualize spatial reconstruction differences.Source: §B.3 · pp. 14–16, Figure 5 · p. 16
  • Analytical and PE-EK-PINN real and imaginary fields retain consistent spatial and oscillatory structures from one dipole through the 16×16 array.Complex-error maps show where reconstruction differences occur.Source: §B.3 · p. 14, Figure 3 · p. 15
  • 26.84 to 29.59 minutes per PE-EK-PINN level as dipoles grow from 4 to 256, versus extrapolated direct PE-PINN times of 17.9 and 71.4 hours for 8 × 8 and 16 × 16 arrays.Direct training time rises approximately linearly with active primitive kernels; cumulative cascade time grows much more slowly.Source: §B.3 · p. 14, Figure 4 · p. 15
  • Cross-array visualizations follow reuse from a 0.5λ line source to single, 2 × 2, and 4 × 4 crosses, with consistent interference patterns and oscillatory structures.Complex-error maps show reconstruction differences at each hierarchy level.Source: Figure 6 · p. 16, §B.3 · pp. 16–17
  • 16.4 minutes for a single cross grows to approximately 32 minutes for 2 × 2 and 4 × 4 stages, while direct PE-PINN rises from 60.8 to 245.1 minutes for single and 2 × 2 crosses.The direct 4 × 4 time is extrapolated to approximately 980 minutes (16.3 hours); cumulative PE-EK-PINN time for 2 × 2 is 91.2 minutes.Source: Figure 7 · p. 17, §B.3 · p. 17
Loading 2609.38023v1…