A holomorphic neural network framework for 3D boundary value problems governed by harmonic potentials
arXiv:2605.31231v1 Announce Type: cross Abstract: We present a neural-network-based framework for the solution of three-dimensional boundary value problems where the solution is expressible in terms of harmonic potentials. The approach leverages the Whittaker integral formula, which allows representing the solution through functions that are holomorphic with respect to a suitable complex variable. These functions are subsequently approximated using holomorphic neural networks, which guaranty fulfillment of the holomorphicity requirement. A key feature of the proposed formulation is that the go
The continuous advancements in neural network architectures and computational methods are enabling novel approaches to traditional mathematical problems, leading to this development leveraging holomorphic networks for complex boundary value problems.
This research provides a new, potentially more accurate and efficient, computational method for solving critical engineering and physics problems, which could accelerate innovation in fields relying on 3D simulations.
The method of approximating solutions for complex 3D boundary value problems is changing from traditional numerical methods to a more specialized neural network approach that guarantees holomorphicity.
- · AI researchers (especially in scientific computing)
- · Engineers (e.g., aerospace, mechanical, civil)
- · Scientific simulation software developers
- · High-performance computing sector
- · Developers of legacy numerical solvers
More accurate and faster simulations for complex physical phenomena become possible.
This could lead to accelerated design cycles for advanced materials, aerospace components, or biological systems.
New industries or products might emerge that were previously computationally infeasible due to the complexity of their underlying physics.
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Read at arXiv cs.LG