Physics-Informed Holomorphic Neural Networks

• Physics-Informed Holomorphic Neural Networks (PIHNNs) are complex-valued models that leverage holomorphic potentials to intrinsically embed elliptic PDE constraints, eliminating interior residual evaluations.
• They transform the learning process into a boundary-only training problem, simplifying optimization by enforcing analytic solution representations through complex-linear layers and holomorphic activations.
• Variants such as PIHKAN demonstrate enhanced accuracy and efficiency on benchmarks including Laplace, Helmholtz, and plane elasticity, even with fewer parameters and reduced training time.

Searching arXiv for papers on PIHNNs and related holomorphic PINN methods. I’ll look for arXiv entries on “Physics-Informed Holomorphic Neural Networks” and closely related holomorphic/complex PINN methods. Physics-Informed Holomorphic Neural Networks (PIHNNs) are physics-informed networks in which the neural architecture is constrained to output holomorphic functions of the complex variable $z=x+iy$. For classes of two-dimensional boundary value problems that admit holomorphic potential representations, this constraint embeds the governing PDE into the model itself: the interior equation is satisfied by construction, and training is reduced to enforcing boundary conditions. PIHNNs therefore form a specialized branch of the broader PINN framework, which ordinarily approximates a real-valued solution by minimizing a loss containing data terms, boundary or initial-condition terms, and a PDE residual evaluated at interior collocation points. A related but distinct development is the complex-analysis-inspired "compleX-PINN", whose activation is derived from Cauchy’s integral formula but whose reported PDE experiments remain real-valued rather than fully holomorphic (Calafà et al., 2024, Calafà et al., 30 Jul 2025, Small, 2023, Si et al., 7 Feb 2025).

1. From PDE-residual PINNs to boundary-only holomorphic learning

In standard PINNs, the PDE is treated as a soft constraint in the loss. A typical formulation minimizes a combination of residual, boundary, and initial terms,

$$\mathcal{L}(\theta)=\lambda_F\mathcal{L}_F(\theta)+\lambda_B\mathcal{L}_B(\theta)+\lambda_I\mathcal{L}_I(\theta),$$

with $\mathcal{L}_F$ computed at interior collocation points and $\mathcal{L}_B,\mathcal{L}_I$ evaluated on the boundary and initial surface. The general PINN viewpoint, as analyzed in the real-valued setting, is that the network is a differentiable surrogate $u_\theta$ and the PDE residual is enforced through automatic differentiation inside the optimization loop (Small, 2023, Si et al., 7 Feb 2025).

PIHNNs replace this soft enforcement by an analytic representation of the solution space. If a PDE admits holomorphic potentials $\varphi_1,\dots,\varphi_{N_*}\in H(\Omega)$ and a fixed reconstruction operator $\mathcal{F}^*$ such that

$$\mathbf{u}=\mathcal{F}^*[\varphi_1,\dots,\varphi_{N_*}],$$

then every admissible network output already lies in $\ker\mathcal{F}$. The learning problem becomes a boundary-only problem:

$$\mathcal{L}[\varphi_1,\dots,\varphi_{N_*}] = \mathbb{E}_{\mathbf{x}\sim\mathcal{P}_{\partial\Omega}} \big[ \boldsymbol{\lambda}_{\mathcal B}\cdot |\mathcal{B}[\mathcal{F}^*[\varphi_1,\dots,\varphi_{N_*}]](\mathbf{x})|^2 \big].$$

This changes both the optimization target and the sampling regime: no interior PDE residual is evaluated, no PDE-versus-BC weight balancing is required, and training points are drawn only from $$\mathcal{L}(\theta)=\lambda_F\mathcal{L}_F(\theta)+\lambda_B\mathcal{L}_B(\theta)+\lambda_I\mathcal{L}_I(\theta),$$0 (Calafà et al., 30 Jul 2025).

Two misconceptions recur in discussions of the subject. The first is that PIHNNs are merely complex-valued PINNs. In the core PIHNN formulation, complex values are not incidental; holomorphicity is the structural mechanism that enforces the PDE. The second is that holomorphicity is imposed through a Cauchy–Riemann penalty term. In the principal PIHNN constructions, holomorphicity is instead enforced by architecture through complex-linear layers and holomorphic activations (Calafà et al., 2024).

2. Holomorphic representations of elliptic PDEs

The PIHNN framework applies when the PDE admits a holomorphic representation on a two-dimensional domain $$\mathcal{L}(\theta)=\lambda_F\mathcal{L}_F(\theta)+\lambda_B\mathcal{L}_B(\theta)+\lambda_I\mathcal{L}_I(\theta),$$1. In the formulation of (Calafà et al., 30 Jul 2025), a problem

$$\mathcal{L}(\theta)=\lambda_F\mathcal{L}_F(\theta)+\lambda_B\mathcal{L}_B(\theta)+\lambda_I\mathcal{L}_I(\theta),$$2

admits a holomorphic representation if

$$\mathcal{L}(\theta)=\lambda_F\mathcal{L}_F(\theta)+\lambda_B\mathcal{L}_B(\theta)+\lambda_I\mathcal{L}_I(\theta),$$3

This places PIHNNs within classical complex analysis rather than generic function approximation.

PDE	$$\mathcal{L}(\theta)=\lambda_F\mathcal{L}_F(\theta)+\lambda_B\mathcal{L}_B(\theta)+\lambda_I\mathcal{L}_I(\theta),$$4	Holomorphic representation
Laplace	1	$$\mathcal{L}(\theta)=\lambda_F\mathcal{L}_F(\theta)+\lambda_B\mathcal{L}_B(\theta)+\lambda_I\mathcal{L}_I(\theta),$$5
Biharmonic	2	$$\mathcal{L}(\theta)=\lambda_F\mathcal{L}_F(\theta)+\lambda_B\mathcal{L}_B(\theta)+\lambda_I\mathcal{L}_I(\theta),$$6
Plane linear elasticity	2	Kolosov–Muskhelishvili representation
Helmholtz	1	$$\mathcal{L}(\theta)=\lambda_F\mathcal{L}_F(\theta)+\lambda_B\mathcal{L}_B(\theta)+\lambda_I\mathcal{L}_I(\theta),$$7

For plane linear elasticity, the representation is especially explicit. With holomorphic potentials $$\mathcal{L}(\theta)=\lambda_F\mathcal{L}_F(\theta)+\lambda_B\mathcal{L}_B(\theta)+\lambda_I\mathcal{L}_I(\theta),$$8, the displacement field can be written as

$$\mathcal{L}(\theta)=\lambda_F\mathcal{L}_F(\theta)+\lambda_B\mathcal{L}_B(\theta)+\lambda_I\mathcal{L}_I(\theta),$$9

with $\mathcal{L}_F$0. In the elasticity-specific PIHNN formulation, stresses and displacements are recovered from two holomorphic potentials through the Kolosov–Muskhelishvili formulas, and the equilibrium equations are then automatically satisfied in the interior (Calafà et al., 2024, Calafà et al., 30 Jul 2025).

For Helmholtz, the representation is not a direct real-part formula but a Vekua transform of a harmonic function. A practical form used in PIHNN/PIHKAN work is

$\mathcal{L}_F$1

where $\mathcal{L}_F$2 is the Bessel function of the first kind of order $\mathcal{L}_F$3. The integral is approximated by Gauss–Legendre quadrature, while holomorphicity remains the mechanism that parameterizes the admissible solution family (Calafà et al., 30 Jul 2025).

This PDE class is specialized. The framework is presented for elliptic problems in 2D, particularly Laplace, biharmonic equations, plane linear elasticity, and Helmholtz. A plausible implication is that the decisive question is not whether a PDE is “complex-valued,” but whether its solution space is accessible through holomorphic potentials and a tractable operator $\mathcal{L}_F$4.

3. Holomorphic architectures, activations, and approximation theory

A PIHNN is a complex-valued feedforward network

$\mathcal{L}_F$5

with complex-linear maps $\mathcal{L}_F$6 and analytic activation $\mathcal{L}_F$7. Because compositions and linear combinations of holomorphic functions are holomorphic, every network realization remains holomorphic on its domain of definition. In the elasticity formulation, the activation finally adopted is $\mathcal{L}_F$8, after discussion of alternatives including $\mathcal{L}_F$9, $\mathcal{L}_B,\mathcal{L}_I$0, and $\mathcal{L}_B,\mathcal{L}_I$1 (Calafà et al., 2024).

The approximation-theoretic basis is unusually strong. For entire non-polynomial activations, shallow holomorphic networks are dense in $\mathcal{L}_B,\mathcal{L}_I$2 under uniform convergence on compact subsets of a simply-connected domain $\mathcal{L}_B,\mathcal{L}_I$3. In the formulation of (Calafà et al., 2024), one-hidden-layer networks

$\mathcal{L}_B,\mathcal{L}_I$4

can approximate any holomorphic $\mathcal{L}_B,\mathcal{L}_I$5 on compact subsets. This is the direct analog, in holomorphic function spaces, of universal approximation results used to motivate real-valued PINNs.

The 2025 extension introduces a holomorphic Kolmogorov–Arnold architecture, PIHKAN, in which trainable functions live on edges rather than being fixed activations at nodes. Each edge map is chosen as a polynomial

$\mathcal{L}_B,\mathcal{L}_I$6

using the monomial basis $\mathcal{L}_B,\mathcal{L}_I$7. Because each edge function is polynomial, the entire network is holomorphic on $\mathcal{L}_B,\mathcal{L}_I$8. PIHKAN is motivated by the Kolmogorov–Arnold representation and is reported to achieve lower or comparable errors with fewer trainable parameters than holomorphic MLPs in the reported elliptic benchmarks (Calafà et al., 30 Jul 2025).

A separate line of work, "Complex Physics-Informed Neural Network," does not build a strict PIHNN but introduces a Cauchy-integral-inspired activation,

$\mathcal{L}_B,\mathcal{L}_I$9

derived from Cauchy’s integral formula. In the reported PDE experiments the network is real-valued, but the activation becomes a rational function

$u_\theta$0

if extended to complex arguments, hence holomorphic away from its poles and globally meromorphic. This suggests a natural bridge between complex-analysis-motivated PINNs and genuinely holomorphic, rational-function PIHNN architectures (Si et al., 7 Feb 2025).

4. Training objectives, automatic differentiation, and initialization

Because the PDE is enforced structurally, PIHNN training focuses on boundary conditions. In the elasticity paper, the basic objective is

$u_\theta$1

with $u_\theta$2 and weights chosen proportional to boundary segment length. Dirichlet and Neumann parts are sampled uniformly on $u_\theta$3 and $u_\theta$4, and symmetry terms are added when symmetry conditions are imposed. This is qualitatively different from standard PINNs, whose loss requires interior collocation points and repeated PDE differentiation at those points (Calafà et al., 2024, Small, 2023).

Automatic differentiation remains central, but its role changes. In standard PINNs, AD computes interior residuals such as $u_\theta$5 or $u_\theta$6 for the PDE term. In PIHNNs, AD primarily computes derivatives of holomorphic potentials needed in the analytic reconstruction formulas, such as $u_\theta$7, $u_\theta$8, and derivatives entering stresses, tractions, or Vekua transforms. The 2025 PIHKAN work explicitly uses Wirtinger derivatives for complex parameters,

$u_\theta$9

while the 2024 elasticity implementation uses PyTorch 2.2.2 with built-in complex automatic differentiation (Calafà et al., 30 Jul 2025, Calafà et al., 2024).

Initialization is not a minor detail in holomorphic networks. For exponential activations, the elasticity paper derives a variance-based complex initialization intended to control forward activations and first-, second-, and third-order derivative variances across layers. A practical prescription is

$\varphi_1,\dots,\varphi_{N_*}\in H(\Omega)$0

with $\varphi_1,\dots,\varphi_{N_*}\in H(\Omega)$1 chosen in a stability interval; experiments reported in the appendix indicate that $\varphi_1,\dots,\varphi_{N_*}\in H(\Omega)$2 provides a favorable trade-off, whereas standard complex Xavier or He schemes tend to produce exploding gradients with the exponential activation in this setting (Calafà et al., 2024).

For PIHKAN, the issue reappears in polynomial form. Assuming complex Gaussian inputs and polynomial edge activations, the proposed initialization is

$\varphi_1,\dots,\varphi_{N_*}\in H(\Omega)$3

with zero biases. The same work also uses an affine complex normalization of the first-layer inputs,

$\varphi_1,\dots,\varphi_{N_*}\in H(\Omega)$4

which is holomorphic and therefore preserves the analytic structure of the network (Calafà et al., 30 Jul 2025).

Sampling can also be adapted to the boundary-only setting. The PIHKAN paper modifies residual-based adaptive distribution (RAD) to resample boundary points according to the boundary residual

$\varphi_1,\dots,\varphi_{N_*}\in H(\Omega)$5

focusing training near corners and other boundary regions with large error (Calafà et al., 30 Jul 2025).

5. Applications, geometries, and reported empirical behavior

The initial PIHNN work centers on plane linear elasticity. Benchmarks include a circular ring under pressure, a square plate with a circular hole under uniaxial tension, a clamped plate under shear, a rail cross-section under compression, and a multiply-connected plate-with-hole problem handled by domain decomposition. The reported behavior is consistent across these examples: accurate interior stress and displacement fields are obtained from small boundary-trained complex networks, while the largest errors appear near holes, sharp corners, or other singular boundary features (Calafà et al., 2024).

The subsequent PIHKAN work broadens the scope to nonhomogeneous Laplace problems on an L-shaped domain, Helmholtz problems on a square, and plane linear elasticity on a square plate with a circular hole. All reference solutions are computed with FEM in FreeFem++, and the comparisons explicitly include standard PINNs, holomorphic MLPs (PIHNN), and holomorphic KANs (PIHKAN) (Calafà et al., 30 Jul 2025).

Benchmark	Reported PINN result	Reported holomorphic result
L-shaped Laplace	526 s, 7851 params, rel. $\varphi_1,\dots,\varphi_{N_*}\in H(\Omega)$6	PIHNN: 13 s, 7622 params, $\varphi_1,\dots,\varphi_{N_*}\in H(\Omega)$7; PIHKAN: 12 s, 4302 params, $\varphi_1,\dots,\varphi_{N_*}\in H(\Omega)$8
Square Helmholtz	534 s, 7851 params, rel. $\varphi_1,\dots,\varphi_{N_*}\in H(\Omega)$9	PIHNN: 13 s, 3522 params, $\mathcal{F}^*$0; PIHKAN: 26 s, 1002 params, $\mathcal{F}^*$1
Clamped plate under shear	PINN-100: 71 s; PINN-1000: 724 s	PIHNN: 73 s; stress errors $\mathcal{F}^*$2, $\mathcal{F}^*$3, $\mathcal{F}^*$4

Beyond these summaries, several detailed benchmarks are reported. In the circular-ring elasticity example, the stress-only PIHNN uses two hidden layers with 10 units per network, 200 training points, and 1000 epochs, with training time approximately 15 seconds on a single CPU. In the square plate with a circular hole, a similarly small network trains in approximately 30 seconds. In the clamped plate under shear, a four-layer, 100-unit PIHNN trained for 1000 epochs is compared with a real-valued SciANN PINN trained on 10,000 points; with comparable training time, the PIHNN is reported to be more accurate, and it remains more accurate than the longer-trained PINN-1000 case (Calafà et al., 2024).

For multiply-connected domains, two strategies are reported. The 2024 elasticity paper uses domain decomposition: the domain is split into simply connected subdomains, each with its own PIHNN, and interface conditions

$\mathcal{F}^*$5

are enforced on internal boundaries. The 2025 work introduces a Laurent-series-based alternative that avoids domain decomposition by representing each potential as

$\mathcal{F}^*$6

In the square plate with circular hole benchmark, the Laurent-based methods L-PIHNN and L-PIHKAN are reported as more accurate than domain-decomposition methods DD-PIHNN and DD-PIHKAN, with L-PIHKAN attaining the lowest reported stress errors while using the fewest parameters among the compared holomorphic variants (Calafà et al., 2024, Calafà et al., 30 Jul 2025).

6. Scope, limitations, and open questions

PIHNNs are powerful only where the analytic structure exists. The framework as developed is restricted to elliptic PDEs in 2D whose solution spaces admit holomorphic representations, classical or Vekua-type. It does not directly cover hyperbolic or parabolic equations, strongly heterogeneous coefficients, or nonlinear PDEs in the reported formulations. The elasticity-specific development also assumes homogeneous, isotropic plane linear elasticity without body forces (Calafà et al., 30 Jul 2025, Calafà et al., 2024).

Another limitation follows directly from holomorphic regularity. Holomorphic functions are analytic, so PIHNNs do not exactly reproduce corner singularities, discontinuous tractions, shocks, or other non-analytic phenomena. In the clamped plate under shear and in the rail cross-section example, the reported boundary and field errors are largest near corners and stress concentrations; the learned solution remains smooth where the exact solution has singular or sharply localized behavior. The same issue appears in the broader PINN literature, where smooth network surrogates tend to smear steep gradients or shock-like structures (Calafà et al., 2024, Small, 2023).

The relation between PIHNNs and other complex-valued architectures is also easily overstated. "compleX-PINN" is explicitly described as not being a fully complex-valued holomorphic network in the strict sense; its contribution is a Cauchy-integral-inspired activation and strong empirical performance in stiff and high-frequency real-valued PINN benchmarks. It is best interpreted as a stepping stone toward PIHNN-type models rather than an instance of the mature PIHNN framework itself (Si et al., 7 Feb 2025).

Open directions are stated clearly in the literature. The PIHKAN work points to broader elliptic PDE classes through additional Vekua operators, better quadrature for those operators, optimized KAN kernels, and extensions toward 3D and time-dependent settings. The compleX-PINN paper identifies explicit complex-valued PDEs and outputs, complex-valued AD and optimizers, pole stability, and formal approximation theory for holomorphic rational networks in the PINN context as natural next steps. Taken together, these directions indicate that PIHNN research is moving along two complementary axes: expansion of the PDE classes for which boundary-only holomorphic learning is available, and refinement of the holomorphic architectures used to parameterize the admissible solution space (Calafà et al., 30 Jul 2025, Si et al., 7 Feb 2025).