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

Physics-informed neural networks (PINNs) can solve wave equations without a traditional mesh, but they often struggle with highly oscillatory fields. The network must discover rapid wave patterns through optimization, and neural networks typically learn low-frequency structure first. This problem is especially serious in practical radiation simulations involving singular sources, absorbing boundaries, and fields spanning many wavelengths. In this paper on PE-EK-PINN, Huiwen Zhang, Feng Ye, and Chu Ma propose a way to reuse learned wave structures across increasingly large systems, reducing the amount of physics that must be learned from scratch.

Why ordinary PINNs struggle with practical waves

A conventional PINN represents a field with a neural network and trains it to reduce several losses: the governing partial differential equation, source or excitation conditions, and boundary conditions. For wave problems, however, this approach leaves the network responsible for discovering the field’s oscillatory behavior from coordinates alone.
The authors argue that common manufactured Helmholtz benchmarks can hide this weakness. Such benchmarks often use smooth, separable solutions over relatively short distances. On one example reported in the paper, CoPINN achieves a relative (L_2) error of (0.00500) on a manufactured solution, but its error rises to (0.99458) on a 2.4 GHz dipole-radiation problem. The latter includes a singular excitation, absorbing truncation, and a field extending over roughly 40 wavelengths. PE-PINN performs much better on that radiation example, with a reported relative error of (0.02136).
PE-PINN improves representation by embedding wave behavior into the architecture. It writes the field as a sum of oscillatory kernels multiplied by learnable, smoother envelopes. For a point-like source, one primitive kernel can take the form of a spherical wave, such as (e^{-jk|\mathbf{x}-\mathbf{x}_m|}). Because the kernel already carries much of the rapid phase variation, the neural network can focus on the remaining smooth changes.
That strategy introduces a new bottleneck. If every elementary source requires (q) kernels, a configuration containing (N) sources requires (qN) active kernels. In hierarchically constructed systems, such as dipole arrays or metasurfaces, the number of elementary units can grow geometrically with the hierarchy depth. The kernel dictionary therefore becomes increasingly expensive to construct, store, and train.

Turning trained fields into reusable kernels

PE-EK-PINN addresses this issue by treating a converged subsystem solution as a new physics-aware kernel. The process is hierarchical:

1. Train a PE-PINN for a small subsystem using primitive analytical kernels. 2. Freeze the resulting field representation after convergence. 3. Promote that frozen field to an evolved kernel. 4. Reuse transformed copies of the evolved kernel to represent a larger configuration. 5. Train only new envelopes and gating functions at the larger level. 6. Freeze the new solution and continue the process.
   The evolved kernel is not merely an analytical wave formula. It is a learned composite representation that encodes the collective behavior of an entire subsystem, including interactions among its lower-level components. At the next level, translated copies of this kernel provide the dominant oscillatory structure.
   This reuse is justified by the translation invariance of the homogeneous Helmholtz equation. If a learned kernel satisfies the Helmholtz equation, translating it produces another field satisfying the same equation. The paper focuses on translations, while noting that the framework can extend to other rigid transformations.
   The larger field is still not obtained by simply adding isolated subsystem solutions. The envelopes remain necessary because neighboring subsystems interact, source constraints must be satisfied, and the outer absorbing boundary applies to the composite field. At every level, the method evaluates the PDE residual on the full current field, so physical consistency is re-enforced rather than assumed to be perfectly inherited from cached kernels.

What the reported results show

The central complexity claim concerns the number of active kernels. Direct PE-PINN uses a kernel for each elementary component, so its active-kernel count grows with system size. In the cascading scheme, each lower-level subsystem is represented by one frozen evolved kernel. At a given level, only the copies needed to assemble that level are active.
The authors state that the peak number of active kernels becomes independent of the total number of elementary units. Under their assumptions, cumulative training cost changes from approximately (O(N)) for direct construction to (O(\log N)) for a recursively organized system. This estimate assumes that the number of epochs per hierarchy level stays roughly constant and that per-iteration cost grows approximately linearly with the number of active kernels.
Experiments cover dipole arrays, composite line-source geometries, and cross arrays. The most prominent example is a 256-dipole array. PE-EK-PINN reportedly trains it in 2 hours 11 minutes, compared with an extrapolated 71 hours for direct PE-PINN. The paper describes this as more than a 30-fold speedup, while also reporting a fourfold reduction in relative (L_2) error for that comparison. Across the evaluated configurations, the method is described as achieving reduced or comparable error while substantially lowering training cost.

What to keep in mind

PE-EK-PINN depends on structured configurations that can be built from transformed copies of smaller subsystems. Its reuse argument is most direct in homogeneous media, where the Helmholtz operator is translation invariant. More complicated settings may require suitable transformations or new kernel constructions.
The approach also does not eliminate training at larger scales. Each hierarchy level still requires learning new envelopes and gates, and the full composite PDE, source, and boundary losses must be optimized. Freezing a lower-level solution compresses previously learned structure, but it may also limit how that structure can adapt to a new environment. The paper’s results support the reported scalability gains on the tested radiation geometries; they do not establish that the same complexity or accuracy behavior will hold for every wave problem.
The broader idea is to match the hierarchy of the representation to the hierarchy of the physical system. Instead of expanding a large array into all of its elementary wave sources, PE-EK-PINN repeatedly packages learned collective behavior into reusable components. That allows architectural physics embedding to retain its advantages while avoiding the rapidly growing kernel dictionary that limits direct PE-PINN.