Enriched Holomorphic Neural Networks for 2D and 3D Laplace Problems
Boundary-value problems for the Laplace equation develop singular behaviour near corners, edges and junctions that smooth approximators resolve poorly. We combine holomorphic neural networks with singular functions derived from the local geometry and boundary conditions.
This paper introduces an enriched holomorphic neural network approach to solve Laplace boundary-value problems with improved accuracy, particularly in domains with geometric singularities.
What happened
Researchers developed a neural network architecture that integrates holomorphic functions with singular basis functions to solve Laplace equations. The method constructs harmonic solutions by design, using boundary data and trainable singular terms. It was validated in 2D and 3D scenarios, showing significant accuracy improvements over existing methods.
Why it matters
Accurate solution of Laplace equations is critical for engineering simulations and physical modeling. The method's ability to handle geometric singularities improves reliability in complex domains like radiation detectors and material science applications.
- Neural networks can be structured to inherently satisfy physical constraints like harmonicity
- Geometry-aware singular functions improve solution accuracy in challenging domains
- Trainable exponents provide flexibility for unknown singularity characteristics
- 3D extension demonstrates scalability of the approach to complex problems
The bigger picture
The approach enables more accurate numerical simulations in domains with sharp corners and edges, which is particularly valuable for engineering analysis and scientific computing where traditional methods struggle with singularities.
The enriched holomorphic neural network method provides a novel approach to solving Laplace equations with superior accuracy, particularly in domains with geometric singularities, through a combination of complex analysis and adaptive singular function enrichment.
Technical details
- Holomorphic neural networks with complex-valued activations
- Singular functions derived from local geometry and boundary conditions
- Harmonic constraint enforced through network architecture
- Trainable exponents in singular expansion terms
- 3D implementation using independent networks along complex null directions
FAQ
What problem does this research address?
The research addresses the challenge of solving Laplace boundary-value problems in domains with geometric singularities (corners, edges) where traditional smooth approximators perform poorly.
How does the method improve upon existing approaches?
The method constructs harmonic solutions by design, eliminating the need for interior residuals. It uses geometry-derived singular functions and trainable exponents to capture complex singular behavior more accurately than conventional methods.
What are the key technical contributions?
The key contributions include a holomorphic network architecture with built-in harmonic constraints, a framework for incorporating geometry-based singular functions, and a 3D extension using complex null-direction networks.
This summary and analysis were generated with AI from the original article and may contain errors. Read the original for the full story.
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Originally published by arXiv cs.LG on Oct 12, 2026.
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