cs.LGOct 28, 2026

A Hybrid Physics-informed holomorphic neural networks Method

Authors: Zhijie Su, Qiaolin He, Zhiping Mao, Shiquan Zhang

Organizations: Department of Mathematics, Sichuan University, No. 24, Section 1, South Yihuan Road, Chengdu, 610065, China · Eastern Institute of Technology, Ningbo, No. 2911, Haijiang Avenue, Zhenhai District, Ningbo, 315200, China

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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.

Figures & tables

 u=up+uh,u=u_{p}+u_{h}, 
Algorithm 1 HPIHNN for nonhomogeneous linear PDEs
Figure 1: HPIHNN Flowchart
ProblemSpectral domainSpectral basisMMHidden layers ×\times widthBoundary pointsEpochs
    HPIHNNPINN  
Poisson[−1,1]2[-1,1]^{2}Fourier204×1284\times 1284×1284\times 12854010,000
Biharmonic[−1,1]2[-1,1]^{2}Fourier201×1281\times 1283×1283\times 128 b20020,000
Stokes[0,1]2[0,1]^{2}Fourier642×1282\times 1282×2562\times 25620020,000
Elasticity ( ν=0.499\nu=0.499 )[0,2π]2[0,2\pi]^{2}Fourier642×1282\times 1282×2562\times 25620040,000
Elasticity: square[−0.5,0.5]2[-0.5,0.5]^{2}Chebyshev a405×325\times 325×1285\times 128 c40010,000
Table 1: Spectral discretization and neural network configurations for the numerical examples.
Figure 2: Evolution of the training loss for the HPIHNN.
Figure 3: Numerical comparison of HPIHNN and PINN for the Poisson problem ( 23 ).
MethodComputational time#parametersRL2\mathrm{RL}_{2}RL∞\mathrm{RL}_{\infty}
HPIHNN57.70 s57.70\,\mathrm{s}49,9213.01⋅10−43.01\cdot 10^{-4}3.54⋅10−33.54\cdot 10^{-3}
PINN90.68 s90.68\,\mathrm{s}50,0491.94⋅10−21.94\cdot 10^{-2}4.40⋅10−24.40\cdot 10^{-2}
Table 2: Accuracy and computational efficiency of HPIHNN and PINN for the Poisson problem ( 23 ).
Figure 4: Numerical comparison of HPIHNN and PINN for the biharmonic problem ( 26 ).
MethodComputational time#parametersRL2\mathrm{RL}_{2}RL∞\mathrm{RL}_{\infty}
HPIHNN87.27 s87.27\,\mathrm{s}7707.93⋅10−57.93\cdot 10^{-5}3.41⋅10−43.41\cdot 10^{-4}
PINN903.68 s903.68\,\mathrm{s}33,5372.64⋅10−12.64\cdot 10^{-1}4.56⋅10−14.56\cdot 10^{-1}
Table 3: Accuracy and computational efficiency of HPIHNN and PINN for the biharmonic problem ( 26 ).
MethodComputational time#parametersError normuuvvpp
HPIHNN61.99 s61.99\,\mathrm{s}33794RL2\mathrm{RL}_{2}6.28⋅10−56.28\cdot 10^{-5}1.50⋅10−41.50\cdot 10^{-4}1.68⋅10−61.68\cdot 10^{-6}
RL∞\mathrm{RL}_{\infty}2.18⋅10−42.18\cdot 10^{-4}3.46⋅10−43.46\cdot 10^{-4}6.02⋅10−66.02\cdot 10^{-6}
PINN150.57 s150.57\,\mathrm{s}67588RL2\mathrm{RL}_{2}1.33⋅10−31.33\cdot 10^{-3}3.24⋅10−33.24\cdot 10^{-3}4.34⋅10−44.34\cdot 10^{-4}
RL∞\mathrm{RL}_{\infty}1.78⋅10−31.78\cdot 10^{-3}3.43⋅10−33.43\cdot 10^{-3}7.48⋅10−47.48\cdot 10^{-4}
Table 4: Accuracy and computational efficiency of HPIHNN and PINN for the Stokes problem ( 28 ).
Figure 5: Numerical comparison of HPIHNN and PINN for the Stokes problem ( 28 ).
MethodComputational time#parametersError normσxx\sigma_{xx}σyy\sigma_{yy}σxy\sigma_{xy}uuvv
HPIHNN111.13 s111.13\,\mathrm{s}33794RL2\mathrm{RL}_{2}5.42⋅10−45.42\cdot 10^{-4}5.44⋅10−45.44\cdot 10^{-4}2.75⋅10−22.75\cdot 10^{-2}7.52⋅10−47.52\cdot 10^{-4}1.23⋅10−31.23\cdot 10^{-3}
RL∞\mathrm{RL}_{\infty}5.20⋅10−45.20\cdot 10^{-4}5.03⋅10−45.03\cdot 10^{-4}2.02⋅10−12.02\cdot 10^{-1}2.27⋅10−32.27\cdot 10^{-3}6.28⋅10−36.28\cdot 10^{-3}
PINN266.86 s266.86\,\mathrm{s}67074RL2\mathrm{RL}_{2}9.01⋅10−19.01\cdot 10^{-1}9.03⋅10−19.03\cdot 10^{-1}5.79⋅10−15.79\cdot 10^{-1}5.70⋅10−15.70\cdot 10^{-1}1.85⋅1001.85\cdot 10^{0}
RL∞\mathrm{RL}_{\infty}7.24⋅10−17.24\cdot 10^{-1}7.26⋅10−17.26\cdot 10^{-1}5.00⋅10−15.00\cdot 10^{-1}5.67⋅10−15.67\cdot 10^{-1}1.92⋅1001.92\cdot 10^{0}
Table 5: Accuracy and computational efficiency of HPIHNN and PINN for the linear elasticity problem ( 29 )–( 31 ) with ν=0.499\nu=0.499 .
Figure 6: Numerical comparison of HPIHNN and PINN for the linear elasticity problem ( 29 )–( 31 ) with ν=0.499\nu=0.499 .
Figure 7: Numerical comparison of HPIHNN and PINN for the linear elasticity problem ( 29 )–( 31 ) with ν=0.499\nu=0.499 .
MethodComputational time#parametersRL2\mathrm{RL}_{2}
   σxx\sigma_{xx}σyy\sigma_{yy}σxy\sigma_{xy}uuvv
HPIHNN255.95 s255.95\,\mathrm{s}17284172845.26⋅10−15.26\cdot 10^{-1}9.31⋅10−29.31\cdot 10^{-2}1.15⋅10−11.15\cdot 10^{-1}1.80⋅10−21.80\cdot 10^{-2}2.72⋅10−22.72\cdot 10^{-2}
PINN332.29 s332.29\,\mathrm{s}66690666907.55⋅10−17.55\cdot 10^{-1}1.30⋅10−11.30\cdot 10^{-1}1.69⋅10−11.69\cdot 10^{-1}2.38⋅10−22.38\cdot 10^{-2}3.76⋅10−23.76\cdot 10^{-2}
Table 6: Accuracy and computational efficiency of HPIHNN and PINN for the first benchmark of the linear elasticity problems in Section 4.4.2 .
Figure 8: Numerical comparison of HPIHNN and PINN for the first benchmark of the linear elasticity problems in Section 4.4.2 .
Figure 9: Geometry of the Rail Problem
Figure 10: Numerical comparison of HPIHNN and PINN for the second benchmark (Rail) of the linear elasticity problems in Section 4.4.2 .
MethodComputational time#parametersRL2\mathrm{RL}_{2}
   σxx\sigma_{xx}σyy\sigma_{yy}σxy\sigma_{xy}uuvv
HPIHNN553.32 s553.32\,\mathrm{s}17284172842.61⋅10−12.61\cdot 10^{-1}7.38⋅10−27.38\cdot 10^{-2}2.37⋅10−12.37\cdot 10^{-1}5.15⋅10−25.15\cdot 10^{-2}6.40⋅10−26.40\cdot 10^{-2}
PINN690.45 s690.45\,\mathrm{s}66690666904.59⋅10−14.59\cdot 10^{-1}3.03⋅10−13.03\cdot 10^{-1}6.38⋅10−16.38\cdot 10^{-1}7.55⋅10−17.55\cdot 10^{-1}3.98⋅10−13.98\cdot 10^{-1}
Table 7: Accuracy and computational efficiency of HPIHNN and PINN for the second benchmark (Rail) of the linear elasticity problems in Section 4.4.2 .
Figure 11: Geometry for the circular-hole stress-concentration problem.
Figure 12: Numerical comparison of HPIHNN and PINN for the third benchmark of the linear elasticity problems in Section 4.4.2 .
MethodComputational time#parametersRL2\mathrm{RL}_{2}
   σxx\sigma_{xx}σyy\sigma_{yy}σxy\sigma_{xy}uuvv
HPIHNN153.38 s153.38\,\mathrm{s}883688361.20⋅10−21.20\cdot 10^{-2}5.68⋅10−35.68\cdot 10^{-3}2.13⋅10−22.13\cdot 10^{-2}2.42⋅10−32.42\cdot 10^{-3}1.88⋅10−31.88\cdot 10^{-3}
PINN243.34 s243.34\,\mathrm{s}33666336661.22⋅10−21.22\cdot 10^{-2}6.66⋅10−36.66\cdot 10^{-3}2.18⋅10−22.18\cdot 10^{-2}2.86⋅10−32.86\cdot 10^{-3}1.89⋅10−31.89\cdot 10^{-3}
Table 8: Accuracy and computational efficiency of HPIHNN and PINN for the third benchmark of the linear elasticity problems in Section 4.4.2 .
Figure 13: Numerical comparison of HPIHNN and PINN for the linear elasticity problem in a heterogeneous domain.
MethodComputational time#parametersRL2\mathrm{RL}_{2}
   σxx\sigma_{xx}σyy\sigma_{yy}σxy\sigma_{xy}uuvv
HPIHNN311.54 s311.54\,\mathrm{s}35344353446.78⋅10−26.78\cdot 10^{-2}2.33⋅10−22.33\cdot 10^{-2}1.06⋅10−11.06\cdot 10^{-1}7.65⋅10−37.65\cdot 10^{-3}5.69⋅10−35.69\cdot 10^{-3}
PINN620.90 s620.90\,\mathrm{s}1346641346646.26⋅10−26.26\cdot 10^{-2}2.28⋅10−22.28\cdot 10^{-2}1.01⋅10−11.01\cdot 10^{-1}3.43⋅10−23.43\cdot 10^{-2}1.23⋅10−21.23\cdot 10^{-2}
Table 9: Accuracy and computational efficiency of HPIHNN and PINN for the linear elasticity problem in a heterogeneous domain.
MethodComputational time#parametersRL2\mathrm{RL}_{2}
HPIHNN42.06 s42.06\,\mathrm{s}499214.90⋅10−34.90\cdot 10^{-3}
PINN60.25 s60.25\,\mathrm{s}500497.70⋅10−17.70\cdot 10^{-1}
Table 10: Accuracy and computational efficiency of HPIHNN and PINN for the Helmholtz problem ( 34 ).
Figure 14: Numerical comparison of HPIHNN and PINN for the Helmholtz problem ( 34 ).
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