Holomorphic Neural Networks With Singular Functions Beat All Methods on Laplace Benchmarks

Neural solvers for partial differential equations keep stumbling over the same things: corners, edges and material junctions. That is exactly where solutions of the Laplace equation stop being smooth, and a network with smooth activations approximates them poorly. A new paper on arXiv (cs.LG, 2610.12541) proposes enriched holomorphic neural networks. According to the authors, the model was the most accurate of the tested methods on every problem. On tests where the enrichment describes the singularity exactly, the margin was several orders of magnitude. On a model of a CdZnTe radiation detector with many singular points, in both 2D and 3D, the advantage over the next-best method was roughly 1.7–2.1 times.

How the model works

The idea rests on two components. The first is a holomorphic neural network. Holomorphic functions in the plane are automatically harmonic, meaning they satisfy the Laplace equation. So there is no need to add an interior residual to the loss function, as classic physics-informed approaches do. Boundary data is enough for the network.

The second component is singular functions, built from the local geometry and boundary conditions. They take over the "sharp" part of the solution near the singularities. The network itself approximates the smooth remainder. The division of labor is simple: the known analytics are built in explicitly, and the neural network learns only what could not be written out by hand.

It is also important that the singularity exponents are learnable. If the expansion near a singular point is not known in advance, for example at a complex material junction, the model fits the exponents from data. Without this, the method would work only on problems with ready-made analytical theory.

The move to 3D

Holomorphy is a property of the complex plane, so the authors needed a workaround for the three-dimensional case. They use independent networks evaluated along complex null directions. The abstract does not disclose implementation details, but the principle is clear: harmonicity is preserved in 3D too, and an interior residual is still not required.

What the abstract shows:

• the model is harmonic by construction, so no interior residual is needed;
• the smooth part is learned from boundary data only;
• the singularity exponents are learned, so knowing the expansion in advance is not necessary;
• 2D and 3D are covered by a single approach;
• the best result on all tested problems;
• a margin of several orders of magnitude where the enrichment describes the singularity exactly;
• a 1.7–2.1 times gain on the CdZnTe detector in both dimensions.

Where this could be useful

The 1.7–2.1 times figure looks more modest than "several orders of magnitude," but it reflects practice more honestly. The CdZnTe detector has many singular points, and the enrichment cannot always describe all of them exactly. The gain holds even then. For engineering calculations of electric fields, this matters: errors at electrode corners and on material boundaries affect the simulation of detector response.

The practical takeaway for scientific ML researchers is this: if a problem has geometric singularities, it is better to build the known mathematics into the architecture than to hope the network learns everything on its own. Teams that compute fields in sensors, microelectronics and electrostatics should test the method on their own geometries. Before moving it into production, they need to wait for the code to be released and for independent reproductions.

For now this is a preprint without peer review, and the abstract does not name the methods it was compared against. The limitation is also obvious: the approach is tailored to the Laplace equation and related harmonic problems, and extending it to more complex equations will require separate work.