cs.LGOct 8, 2026

Enriched Holomorphic Neural Networks for 2D and 3D Laplace Problems

Authors: Kalp Shah, Andrei Vlad Nicula, Mihnea Andrei Bârsan, Robert-Stefan Sofroni, Agneya Bhardwaj, Jan Jelínek, Alejandro Valverde Mahou

Organizations: University of Amsterdam · Technical University of Denmark

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Abstract

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. The representation is harmonic by construction, so no interior residual is needed: the network fits the smooth remainder from boundary data alone, and the enrichment terms capture the singular behaviour. Singular exponents are trainable, so the construction also applies when the local expansion is not known in advance. We extend the approach to three dimensions using independent networks evaluated along complex null directions. The enriched model is the most accurate tested method on every problem. Its margin is several orders of magnitude where the enrichment captures the singularity exactly, and a factor of about 1.7-2.1 over the next-best method on a CdZnTe radiation detector with many singular sites, in both two and three dimensions.

Figures & tables

Figure 1: Singular behaviour at a D →\to N junction with ω=2π/3\omega=2\pi/3 , λ=3/4\lambda=3/4 . Left: the leading mode rλsin⁡(λθ)r^{\lambda}\sin(\lambda\theta) is bounded at the vertex. Right: the gradient magnitude ∣∇u∣∼rλ−1=r−1/4|\nabla u|\sim r^{\lambda-1}=r^{-1/4} diverges there. The enrichment term wλw^{\lambda} represents this mode directly.
JunctionLocal solution∣∇u∣|\nabla u|Term EλE_{\lambda}
Dirichlet to Neumannrλsin⁡(λθ)r^{\lambda}\sin(\lambda\theta) , λ=(2k−1)π/(2ω)\lambda=(2k-1)\pi/(2\omega)rλ−1r^{\lambda-1}wλw^{\lambda}
Both Dirichlet, value jump Δg\Delta g(Δg/ω) θ(\Delta g/\omega)\,\thetar−1r^{-1}arg⁡w\arg w
Both Neumann, flux jump, ω∈πN\omega\in\pi\mathbb{N}ℜ[wLog⁡w]\Re[w\operatorname{Log}w]∣ln⁡r∣|\ln r|wLog⁡ww\operatorname{Log}w
Both Neumann, equal flux, ω>π\omega>\pirλcos⁡(λθ)r^{\lambda}\cos(\lambda\theta) , λ=kπ/ω\lambda=k\pi/\omegarλ−1r^{\lambda-1}wλw^{\lambda}
Both Dirichlet, equal values, ω>π\omega>\pirλsin⁡(λθ)r^{\lambda}\sin(\lambda\theta) , λ=kπ/ω\lambda=k\pi/\omegarλ−1r^{\lambda-1}wλw^{\lambda}
OtherwiseC∞C^{\infty} away from vertexboundednone
Table 1: Enrichment dictionary. Each junction matches one row and contributes the term EλE_{\lambda} with a trainable coefficient cqc_{q} . Local coordinates are centred at the junction zcz_{c} , with w=z−zc=reiθw=z-z_{c}=re^{i\theta} ; ω\omega is the interior angle and k=1,2,…k=1,2,\dots indexes the ladder. Every term is holomorphic, so the representation remains exactly harmonic. Figure 5 is a visualization of this table.
Figure 2: Forward pass of the 3D model. The point is projected onto KK complex coordinates ζk=ak⋅x^+bk\zeta_{k}=a_{k}\cdot\hat{\bm{x}}+b_{k} along null directions, each coordinate is passed through its own holomorphic network, and the outputs are summed. The enrichment terms are evaluated at the physical point and added before the real part is taken.
ProblemMLP PINNPlainEnriched
Closed-form solution
Sector 3π/23\pi/29.5×10−29.5\times 10^{-2}7.7×10−47.7\times 10^{-4}3.3×10−12\mathbf{3.3\times 10^{-12}}
Sector 7π/47\pi/41.8×10−11.8\times 10^{-1}1.9×10−21.9\times 10^{-2}1.4×10−16\mathbf{1.4\times 10^{-16}}
L-shape7.4×10−27.4\times 10^{-2}7.5×10−47.5\times 10^{-4}2.9×10−16\mathbf{2.9\times 10^{-16}}
Slit disk1.8×10−11.8\times 10^{-1}3.2×10−13.2\times 10^{-1}2.8×10−12\mathbf{2.8\times 10^{-12}}
Series reference
Table 2: Errors in 2D. All entries report ϵu\epsilon_{u} , median over three seeds at the best budget of Appendix F , chosen by the same rule for every method. Plain and Enriched are the same network without and with the terms of Table 1 .
Figure 3: Enriched model on the 2D CZT detector. Left: normalized potential. Right: error against the reference
ProblemPINNDeep RitzWANBallini et al.Enriched
Closed-form solution
L-prism8.7×10−48.7\times 10^{-4}2.7×10−32.7\times 10^{-3}5.6×10−35.6\times 10^{-3}1.3×10−31.3\times 10^{-3}6.9×10−9\mathbf{6.9\times 10^{-9}}
uAu_{A}8.9×10−48.9\times 10^{-4}4.5×10−34.5\times 10^{-3}1.2×10−21.2\times 10^{-2}1.9×10−31.9\times 10^{-3}1.8×10−4\mathbf{1.8\times 10^{-4}}
uBu_{B}1.1×10−31.1\times 10^{-3}5.9×10−35.9\times 10^{-3}1.2×10−21.2\times 10^{-2}1.6×10−31.6\times 10^{-3}1.6×10−4\mathbf{1.6\times 10^{-4}}
Smooth cube1.5×10−51.5\times 10^{-5}5.9×10−45.9\times 10^{-4}2.2×10−32.2\times 10^{-3}2.0×10−42.0\times 10^{-4}4.6×10−6\mathbf{4.6\times 10^{-6}}
Series reference
Table 3: Relative error ϵu\epsilon_{u} in 3D, median over three converged seeds at each method’s best schedule. The detector is scored against the finite-element reference. ‡ Extension of WAN to mixed boundary data.
Figure 4: Enriched model on the 3D CZT detector, plane z=21z=21 mm. Left: normalized potential. Right: error against the reference, scaled by the root mean square of the reference solution.
Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Enrichment terms and branch cuts for the rows of Table 1 . (a) A Dirichlet to Neumann junction gives wλw^{\lambda} . (b) A Dirichlet value jump gives arg⁡w\arg w , which carries the jump. (c) A Neumann flux jump gives wLog⁡ww\operatorname{Log}w . (d) A re-entrant corner with equal Dirichlet values gives wλw^{\lambda} ; the equal-flux Neumann row has the same term. Each cut leaves the site along the exterior direction.
ProblemDomainλ\lambdaReference
Sector 3π/23\pi/2r<1r<1 , 0<θ<3π/20<\theta<3\pi/22/32/3exact
Sector 7π/47\pi/4r<1r<1 , 0<θ<7π/40<\theta<7\pi/44/74/7exact
L-shape(−1,1)2∖[0,1]×[−1,0](-1,1)^{2}\setminus[0,1]\times[-1,0]2/32/3exact
Slit diskr<1r<1 cut along 0≤x<10\leq x<1 , y=0y=01/21/2exact
Motz(−1,1)×(0,1)(-1,1)\times(0,1)1/21/2D0=401.162453745234416D_{0}=401.162453745234416
Cracked beam(−1,1)×(0,1)(-1,1)\times(0,1)1/21/2D0=540.56512271362749D_{0}=540.56512271362749
Appendix
Table 4: The named benchmarks. λ\lambda is the leading exponent at the singular site. The reference coefficients are from Lu et al. (2004) .
 MotzCracked beam
nnϵu\epsilon_{u}D0D_{0} errorϵu\epsilon_{u}D0D_{0} error
11.9×10−11.9\times 10^{-1}7.9×10−27.9\times 10^{-2}3.0×10−13.0\times 10^{-1}1.5×10−11.5\times 10^{-1}
45.5×10−35.5\times 10^{-3}4.4×10−34.4\times 10^{-3}2.2×10−22.2\times 10^{-2}1.1×10−21.1\times 10^{-2}
81.9×10−41.9\times 10^{-4}3.5×10−53.5\times 10^{-5}4.4×10−44.4\times 10^{-4}8.6×10−68.6\times 10^{-6}
121.6×10−51.6\times 10^{-5}5.1×10−75.1\times 10^{-7}2.6×10−52.6\times 10^{-5}2.8×10−72.8\times 10^{-7}
Appendix
Table 5: Least-squares fit of the first nn singular terms without the network: ϵu\epsilon_{u} on 20,000 random interior points and the relative error of D0D_{0} . The model uses n=4n=4 .
Figure 6: The six benchmarks of Table 4 , with the exact solution drawn where one exists.
ModelMSERMSE
MLP PINN2.40×10−12.40\times 10^{-1}4.90×10−14.90\times 10^{-1}
Plain1.18×10−41.18\times 10^{-4}1.09×10−21.09\times 10^{-2}
Enriched4.19×10−5\mathbf{4.19\times 10^{-5}}6.47×10−3\mathbf{6.47\times 10^{-3}}
Appendix
Table 6: Potential errors on the CZT cross-section, in units of 350 V, median over three seeds.
Figure 7: The 24 instances of the sampled suite. Dirichlet segments are coloured by their prescribed value, and Neumann segments are dashed. Markers show the sites the dictionary emits, with the shape giving the row of Table 1 .
ConfigurationBoundary lossRatioConvergedTime (s)
starting configuration6.4×10−56.4\times 10^{-5}1.003/18402
depth 23.3×10−63.3\times 10^{-6}0.037/18280
learning rate 0.018.1×10−68.1\times 10^{-6}0.147/18381
Adam steps 40 0001.4×10−51.4\times 10^{-5}0.415/18644
width 161.4×10−51.4\times 10^{-5}0.427/18275
points per segment 1 0248.5×10−68.5\times 10^{-6}0.436/18645
Appendix
Table 7: Ablation on the tuning suite (six instances, three seeds, 18 runs per row), changing one factor at a time from a starting configuration of width 32, depth 3, learning rate 0.003, 20 000 Adam steps, 6 000 L-BFGS steps, 512 points per segment, ladder depth 1 and fixed exponents. Ratio is the median boundary loss relative to the starting configuration; below one is an improvement. Converged counts runs with boundary loss below 10−610^{-6} . The configuration of Section 4 combines the five best changes.
Figure 8: The plain and enriched models across the budget ladder on the named benchmarks, median over three seeds with the spread shaded.
Figure 9: Plain against enriched over the sampled suite, one point per instance and seed. Points below the diagonal improved with enrichment; annotations give the median paired ratio.
Terms∣λ1−λ⋆∣|\lambda_{1}-\lambda^{\star}|Smallest gapSpurious ∣cq∣|c_{q}|ϵu\epsilon_{u}
14.6×10−114.6\times 10^{-11}–05.7×10−115.7\times 10^{-11}
22.3×10−102.3\times 10^{-10}6.3×10−16.3\times 10^{-1}1.9×10−91.9\times 10^{-9}3.4×10−113.4\times 10^{-11}
37.4×10−107.4\times 10^{-10}9.5×10−89.5\times 10^{-8}3.7×10−23.7\times 10^{-2}2.2×10−112.2\times 10^{-11}
48.8×10−108.8\times 10^{-10}1.0×10−71.0\times 10^{-7}1.8×10−11.8\times 10^{-1}3.0×10−113.0\times 10^{-11}
62.0×10−82.0\times 10^{-8}2.5×10−62.5\times 10^{-6}1.0×10−11.0\times 10^{-1}1.3×10−101.3\times 10^{-10}
81.9×10−91.9\times 10^{-9}4.7×10−74.7\times 10^{-7}2.2×10−12.2\times 10^{-1}1.1×10−101.1\times 10^{-10}
Appendix
Table 8: Ladder depth on the 7π/47\pi/4 sector, λ⋆=4/7\lambda^{\star}=4/7 . λ1\lambda_{1} is the trained exponent closest to λ⋆\lambda^{\star} , the gap is the smallest separation between trained exponents, and the spurious amplitude is the largest ∣cq∣|c_{q}| on a non-leading term.
SettingValue
Branches / hidden layers / width16/2/816/2/8
Adam steps120,000, learning rate 0.010.01 , cosine decay
Maximum L-BFGS iterations4,000
Uniform boundary samples1,536 per face (1,024 on the detector)
Near-site samples128 per site, 10−410^{-4} to 10−110^{-1} of the local length
Restarts3 per seed, selected on a fresh boundary sample
Appendix
Table 9: Settings of the enriched model in 3D.
ProblemSeed 0Seed 1Seed 2
L-prism1.3×10−81.3\times 10^{-8}1.7×10−91.7\times 10^{-9}6.9×10−96.9\times 10^{-9}
uAu_{A}1.8×10−41.8\times 10^{-4}1.9×10−41.9\times 10^{-4}1.6×10−41.6\times 10^{-4}
uBu_{B}1.8×10−41.8\times 10^{-4}1.6×10−41.6\times 10^{-4}1.6×10−41.6\times 10^{-4}
Smooth cube4.4×10−64.4\times 10^{-6}4.9×10−64.9\times 10^{-6}4.6×10−64.6\times 10^{-6}
Conical notch4.5×10−94.5\times 10^{-9}6.0×10−96.0\times 10^{-9}6.1×10−96.1\times 10^{-9}
Extruded Motz3.4×10−63.4\times 10^{-6}2.1×10−62.1\times 10^{-6}4.6×10−64.6\times 10^{-6}
Appendix
Table 10: Per-seed relative error ϵu\epsilon_{u} of the enriched model in Table 3 .
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