Kolosov-Muskhelishvili Informed Neural Network

• KMINN is a physics-informed deep learning framework that embeds the Kolosov–Muskhelishvili complex-variable formulation to enforce equilibrium and compatibility laws.
• The model integrates holomorphic neural network architectures with Williams enrichment to accurately capture crack-tip singularities and extract stress intensity factors with sub-percent error.
• A transfer learning strategy in KMINNs reduces simulation time by over 70%, offering a significant speed advantage over conventional PINNs and finite element methods.

Kolosov-Muskhelishvili Informed Neural Network (KMINN) is a specialized class of physics-informed deep learning frameworks for modeling two-dimensional linear elasticity and fracture propagation. By embedding the Kolosov–Muskhelishvili (KM) complex-variable representation into a holomorphic neural network architecture, KMINNs enable mesh-free computation with boundary-only collocation and exact satisfaction of equilibrium and compatibility equations. The approach is particularly effective in fracture mechanics, where crack-tip singularities necessitate specialized enrichment strategies. KMINNs combine classical analytical enrichments (e.g., Williams series) with modern transfer learning to achieve high accuracy in stress intensity factor (SIF) extraction and efficient simulation of crack evolution, yielding sub-percent error and significant speed gains over conventional PINNs and finite element methods (Zhou et al., 1 Jan 2026, Calafà et al., 2024).

1. Mathematical Foundation: Kolosov–Muskhelishvili Representation

KMINNs leverage the KM formalism, in which the field variables in planar elasticity are expressed in terms of two holomorphic functions, $\Phi(z)$ and $\Psi(z)$, with $z = x + i y$. This representation ensures that the governing equations—the strong-form PDEs for stress equilibrium and strain compatibility—are satisfied by construction provided that $\Phi(z)$ and $\Psi(z)$ are holomorphic. The mappings from the KM potentials to physical observables are:

• Stress fields:

$$\sigma_{xx} + \sigma_{yy} = 4\,\mathrm{Re}[\Phi'(z)]$$

$$\sigma_{xx} - \sigma_{yy} + 2i \sigma_{xy} = 2 [\bar{z} \Phi''(z) + \Psi'(z)]$$

• Displacement field:

$u(z) = \kappa \Phi(z) - \bar{z} \Phi'(z) - \Psi(z)$

with $\kappa = 3 - 4\nu$ (plane strain) or $(3 - \nu)/(1+\nu)$ (plane stress), and $\Psi(z)$0 being Poisson's ratio.

All compatibility and equilibrium relations are thus enforced by the holomorphicity conditions $\Psi(z)$1 (Zhou et al., 1 Jan 2026, Calafà et al., 2024).

2. Neural Network Architecture and Enforcement of Holomorphicity

The canonical KMINN architecture comprises two complex-valued multilayer perceptrons (MLPs), one each for $\Psi(z)$2 and $\Psi(z)$3. At each layer, a $\Psi(z)$4-linear map (complex-valued weights and biases) is followed by an entire (holomorphic) activation, typically the exponential function $\Psi(z)$5:

$\Psi(z)$6

This construction ensures that the output is holomorphic by composition. Complex He-style initialization is employed to stabilize the norm of activations and their derivatives; each layer $\Psi(z)$7 uses

$\Psi(z)$8

with layerwise variance $\Psi(z)$9 specifically tuned ("pre-stabilization") (Zhou et al., 1 Jan 2026, Calafà et al., 2024).

A universal approximation theorem for entire neural networks guarantees representation power: for any holomorphic $z = x + i y$0 and desired tolerance $z = x + i y$1, there exists a holomorphic MLP with exponential activation that approximates $z = x + i y$2 to accuracy $z = x + i y$3 on compacta in simply-connected domains (Calafà et al., 2024).

3. Handling Crack-Tip Singularities: Williams Enrichment

Fracture mechanics problems require accurate modeling of the singular stress fields near crack tips. KMINN employs Williams enrichment, directly incorporating the leading term of the Williams series into the neural ansatz: $z = x + i y$4 where $z = x + i y$5 is the crack tip position and $z = x + i y$6 the complex SIF (modes I and II combined). For single-valuedness of the complex square root, the domain is partitioned into subdomains $z = x + i y$7 and $z = x + i y$8, with separate enrichments applied on each. The full neural potentials are then

$z = x + i y$9

This enables automated capture of the $\Phi(z)$0 singularity and accurate extraction of SIFs using path-independent integrals (Zhou et al., 1 Jan 2026).

4. Training Objectives and Boundary-Only Collocation

KMINN training is conducted using only boundary collocation points. The total loss is the sum of mean-square errors on Dirichlet (displacement) boundaries $\Phi(z)$1, Neumann (traction) boundaries $\Phi(z)$2, and interface-matching terms (in domain-decomposition settings): $\Phi(z)$3

$\Phi(z)$4

Interface terms enforce continuity for multiply-connected or domain-decomposed problems. Reference traction and length scales are used to normalize residuals and keep loss components balanced (Zhou et al., 1 Jan 2026, Calafà et al., 2024).

Table 1: Comparison of PINN and KMINN/PIHNN Loss Function Features

Approach	Domain sampling	Governing equations
PINN	Interior + boundary	PDEs and BCs (soft/costly)
KMINN/PIHNN	Boundary only	PDEs satisfied by construction

5. Transfer Learning for Crack Propagation

In incremental crack-growth simulations, KMINNs use transfer learning to exploit the similarity between successive propagation steps. The process is:

1. Extend the crack by $\Phi(z)$5 in the direction predicted from the previous step.
2. Map the SIFs to the new configuration via first-order Cotterell–Rice mapping.
3. Re-use the previous network weights and Williams enrichment as initialization.
4. Fine-tune for a small number of iterations (e.g., 500 Adam + 500 L-BFGS), rather than training from scratch.

This transfer learning approach reduces simulation wall time by over 70% (from 80–140 min to 20–35 min for 50-step crack paths in benchmark problems), with no loss in accuracy (Zhou et al., 1 Jan 2026).

Crack path updates are decided by post-processing the extracted SIFs using any of three classical criteria:

• Maximum Tangential Stress (MTS): crack deflection chosen to maximize $\Phi(z)$6 near the tip.
• Maximum Energy Release Rate (MERR): path selected to maximize $\Phi(z)$7.
• Principle of Local Symmetry (PLS): enforce $\Phi(z)$8 in the local frame.

The various propagation criteria yield nearly identical predicted trajectories and require no modification of the KMINN training objective (Zhou et al., 1 Jan 2026).

6. Numerical Performance and Sampling Properties

KMINN achieves high accuracy on fracture mechanics benchmarks:

• Relative errors in SIF extraction below 1% and $\Phi(z)$9 across mode I, mode II, and mixed-mode configurations.
• No interior sampling or mesh refinement is required near crack tips; $\Psi(z)$0 boundary points suffice for robust learning.
• In direct comparison, KMINNs are mesh-free, physically consistent, and avoid loss-balancing problems inherent to standard PINNs (Zhou et al., 1 Jan 2026, Calafà et al., 2024).

Benchmark problems include center-crack-tension (CCT), center-crack-shear (CCS), and oblique-crack-tension in finite plates, with results closely matching analytical and finite element references even in the presence of strong singularities.

7. Extensions and Theoretical Guarantees

KMINNs and related methods (Physics-Informed Holomorphic Neural Networks, PIHNNs) extend directly to multiply-connected geometries using domain decomposition, with additional boundary/interface losses enforcing continuity. The holomorphic MLP approach, together with tailored weight initialization, mitigates vanishing/exploding gradient issues and enables rapid convergence (Calafà et al., 2024).

Summary of key advantages:

• Exact satisfaction of strong-form elasticity PDEs by construction.
• Mesh-free, boundary-only training enables reduction of computational cost.
• Williams enrichment and holomorphicity automate the capture of singularities, eliminating need for local mesh refinement.
• Transfer learning framework enables efficient crack propagation simulation, reducing cumulative training time by more than 70%.
• Universal approximation guarantees and favorable empirical accuracy (Zhou et al., 1 Jan 2026, Calafà et al., 2024).