Hybrid neural network method improves solving complex physics problems
A Hybrid Physics-informed holomorphic neural networks Method
Machine Learning
Summary
When solving certain complex physics problems using computers, it’s hard to handle equations that don’t fit easily into existing methods. The authors combined two techniques: one that breaks down the problem into simpler parts and another that uses a special type of neural network related to complex math. Their approach avoids complicated mesh setups and works well on tricky shapes and conditions. Tests show their method is often faster and more accurate than standard approaches.
What this means in practice
- •For engineering simulation teams: Simulate mechanical and fluid problems on complex shapes without creating detailed meshes, speeding up solution times and improving accuracy.
- •For software developers in computational physics: Implement hybrid neural network solvers combining spectral methods and physics-informed neural nets to handle nonhomogeneous PDEs efficiently on complex domains.
Authors
Zhijie Su, Qiaolin He, Zhiping Mao, Shiquan Zhang
Abstract
Physics-informed holomorphic neural networks (PIHNNs) exploit complex-analytic representations to satisfy homogeneous PDEs by construction, but their extension to nonhomogeneous problems is nontrivial. We propose a hybrid physics-informed holomorphic neural network (HPIHNN) framework based on a decomposition of the solution into particular and homogeneous components. The particular component is computed by a spectral method on a simple extended domain, while the homogeneous component is approximated by a PIHNN trained only on the boundary with modified boundary data. The resulting formulation avoids meshing complex geometries and retains the dimensionality-reduction advantage of holomorphic representations. The method is applied to nonhomogeneous Poisson, Stokes, biharmonic, linear elasticity, and Helmholtz equations. Numerical experiments covering complex geometries, corner singularities, and nearly incompressible regimes demonstrate that the proposed method accurately resolves the corresponding solutions and, in the considered tests, generally achieves lower errors and shorter training times than standard PINNs. The results indicate that coupling spectral particular solutions with holomorphic neural approximations provides an efficient approach to nonhomogeneous PDEs on complex domains.