Hybrid Parallel KAN/MLP PINNs

• The paper demonstrates that HPKM-PINN significantly reduces L2 error—up to two orders of magnitude—compared to standard PINN or KAN-only models.
• It employs a parallel hybrid strategy that blends KAN’s high-frequency resolution with MLP’s nonlinear expressivity via adaptive convex fusion.
• Domain decomposition and adaptive weighting enhance scalability and robustness, making HPKM-PINN ideal for multiscale and high-dimensional PDE challenges.

The Hybrid Parallel Kolmogorov–Arnold/MLP Physics-Informed Neural Network (HPKM-PINN) is a neural architecture that combines the explicit functional representation capabilities of Kolmogorov–Arnold Networks (KANs) with the spectral and nonlinear expressivity of standard Multi-Layer Perceptrons (MLPs) within the Physics-Informed Neural Networks (PINN) framework. Designed for the efficient and robust solution of partial differential equations (PDEs) and multi-frequency regression tasks, HPKM-PINN utilizes a parallel hybrid strategy and, in recent extensions, adaptive convex weighting and overlapping domain decomposition for improved accuracy and scalability (Xu et al., 30 Mar 2025, Huang et al., 14 Nov 2025).

1. Architecture and Mathematical Formalism

The canonical HPKM-PINN architecture consists of two parallel branches: a KAN branch and an MLP branch, both operating on the same input features (e.g., spatial and temporal coordinates).

• Kolmogorov–Arnold Network (KAN) Branch: Implements the Kolmogorov–Arnold representation theorem, which ensures that any continuous multivariate function on a compact domain can be decomposed as

$$f(x_1,\dots,x_n) = \sum_{q=1}^{2n+1} \Phi_q\left( \sum_{p=1}^n \phi_{q,p}(x_p) \right),$$

with trainable univariate activations $\phi_{q,p}$ and $\Phi_q$. In practical implementations, each $\phi$ and $\Phi$ is parameterized via basis expansions, for instance, employing B-splines and smooth nonlinearities. Alternative parameterizations, such as truncated Fourier series (Fourier-KAN), are employed in high-frequency settings (Huang et al., 14 Nov 2025).

• Multi-Layer Perceptron (MLP) Branch: Constitutes a conventional feedforward architecture,

$$u_{\mathrm{MLP}}(x;\theta) = \Phi^L \circ \cdots \circ \Phi^1(x), \qquad \Phi^{(\ell)}(z) = \sigma(W^{(\ell)}z + b^{(\ell)})$$

with widths and depths tailored to the problem’s dimension and complexity, typically using activation functions such as $\tanh$ or ReLU.

• Hybrid Parallel Fusion: The outputs of the KAN and MLP branches, $u_{\mathrm{KAN}}(x)$ and $u_{\mathrm{MLP}}(x)$, are fused via a convex combination governed by a scalar parameter $\xi$:

$$u_{\mathrm{HPKM}}(x) = \xi\,u_{\mathrm{KAN}}(x) + (1-\xi)\,u_{\mathrm{MLP}}(x), \qquad \xi \in [0, 1].$$

Enhanced variants introduce a trainable scalar $\alpha$ and a monotonic mapping $S(\alpha)\in(0,1)$ (e.g., sigmoid or tanh) for adaptive fusion during training:

$$u_{\mathrm{MHPKM}}(x) = S(\alpha)\,u_{\mathrm{KAN}}(x) + [1-S(\alpha)]\,u_{\mathrm{MLP}}(x).$$

This permits the weighting between KAN and MLP to adapt dynamically, yielding better balancing of frequency components and feature complexity (Huang et al., 14 Nov 2025).

2. Loss Construction and Optimization Protocols

HPKM-PINN adheres to the standard PINN composite loss paradigm for training:

• PDE Loss (Collocation/Residual Term): For a PDE of the form $u_t + \mathcal{N}[u] = 0$ with initial/boundary conditions, the residual at each collocation point is defined as

$$\mathcal{R}(x,t;\theta) = u_t(x,t;\theta) + \mathcal{N}[u(x,t;\theta)],$$

with pointwise losses

$$L_{\mathrm{PDE}} = \frac{1}{N_R} \sum_{i=1}^{N_R} |\mathcal{R}(x_R^i,t_R^i;\theta)|^2\,.$$

• Initial and Boundary Conditions: Losses are imposed by

$$L_{\mathrm{IC}} = \frac{1}{N_{\mathrm{IC}}} \sum_{i=1}^{N_{\mathrm{IC}}} |u(x^i_{\mathrm{IC}},0;\theta) - g(x^i_{\mathrm{IC}})|^2\,, \qquad L_{\mathrm{BC}} = \frac{1}{N_{\mathrm{BC}}} \sum_{i=1}^{N_{\mathrm{BC}}} |B[u](x^i_{\mathrm{BC}},t^i_{\mathrm{BC}};\theta)|^2\,.$$

• Total Loss Function:

$$\mathcal{L}(\theta) = \lambda_{\mathrm{IC}} L_{\mathrm{IC}} + \lambda_{\mathrm{BC}} L_{\mathrm{BC}} + \lambda_{\mathrm{PDE}} L_{\mathrm{PDE}},$$

with $\lambda_{\mathrm{IC}}$, $\lambda_{\mathrm{BC}}$, and $\lambda_{\mathrm{PDE}}$ typically assigned weight 1, or tuned via grid search.

• Optimization: Training is conducted using Adam, with learning rates typically $10^{-2}$ for regression tasks and $10^{-3}$ for PDE-solving, sometimes incorporating learning-rate scheduling. Collocation points are sampled using Sobol or Latin-Hypercube sequences to ensure robust residual evaluation (Xu et al., 30 Mar 2025).
• Domain Decomposition Extension: The modified HPKM-PINN introduces overlapping domain decomposition, partitioning the domain $\Omega$ into overlapping subdomains $(\Omega_i)$ and assembling the global solution via partition-of-unity weighting. Each subdomain is equipped with a local MHPKM network and its own trainable fusion coefficient $\alpha^{(i)}$. The global output is assembled as

$$u(x;\theta) = \sum_{i=1}^N \omega_i(x)\,\left[ S(\alpha^{(i)})\,u^{(i)}_{\mathrm{KAN}}(x) + (1-S(\alpha^{(i)}))\,u^{(i)}_{\mathrm{MLP}}(x) \right],$$

where $\omega_i(x)$ forms a partition of unity (Huang et al., 14 Nov 2025).

3. Benchmark Problems and Empirical Results

HPKM-PINN and its variants have been evaluated on a suite of canonical PDEs and mixed-frequency regression benchmarks:

• Function Fitting (Mixed High–Low Frequency): For $y(x)$ on $x\in[-3,3]$ with high- and low-frequency components, HPKM-PINN (optimal $\xi \approx 0.9$) achieved $L_2$ error of $0.62\%$, outperforming pure MLP ($3.31\%$) and KAN ($4.45\%$).
• 1D Poisson Equation: For $-\frac{d^2u}{dx^2}=f(x),\ u(\pm2\sqrt{\pi})=0,$ with exact $u(x)=\sin(x^2)$,
  • PINN ($\xi=0$): $L_2=1.84\%$,
  • PIKAN ($\xi=1$): $2.31\%$,
  • HPKM ($\xi=0.3$): $0.29\%$.
• 1D Advection, Convection–Diffusion: For $u_t+u_x=0$ and $u_t+4u_x=0.05\,u_{xx}$, respectively, HPKM consistently found lower errors and faster loss convergence at problem-specific $\xi$ (typically $0.2$–$0.7$).
• Helmholtz Equation (2D): For $u_{xx}+u_{yy}+k^2u=q(x,y)$ with $u=0$ Dirichlet boundary, HPKM with $\xi=0.9$ attained lowest $L_2$ error ($0.23$) among compared baselines.
• High-Frequency, High-Dimensional, and Multiscale Benchmarks: The modified HPKM-PINN with overlapping domain decomposition demonstrated marked improvements in high-frequency and high-dimensional settings. For $2$D Helmholtz with $\omega=48$, the method yielded $L_2=8.70\times 10^{-4}$, compared to failure of MLP (error $2.71$) and moderate accuracy of KAN ($1.12\times 10^{-3}$). For $5$D Poisson, comparable $L_2$ ($10^{-4}$) was achieved with concurrent reduction in per-network parameter count (Huang et al., 14 Nov 2025).

Empirical performance summary

PDE / Task	Model	Param Count	Rel. $L_2$ Error	Wall Clock (s)
1D Poisson	PINN [1,20,20,1]	481	$1.84\times 10^{-2}$	88.1
	PIKAN [1,30,30,1]	9600	$2.31\times 10^{-2}$	696.4
	HPKM-PINN	10481	$2.91\times 10^{-3}$	858.7
1D Advection	PINN [2,20,20,20,1]	921	$2.11\times 10^{-3}$	51.5
	PIKAN [2,5,5,1]	400	$5.60\times 10^{-4}$	283.6
	HPKM-PINN	1321	$2.75\times 10^{-4}$	353.6

These results highlight consistent relative error reductions—up to two orders of magnitude—by hybridizing KAN and MLP outputs.

4. Advantages, Limitations, and Algorithmic Insights

Principal advantages documented include significantly reduced approximation error (up to two orders of magnitude over standard PINN or KAN-only PINN), accelerated convergence to low-loss regimes, and superior robustness against data noise, especially under Gaussian perturbations. HPKM-PINN’s fusion parameter $\xi$ furnishes a practical mechanism to interpolate between the spectral coverage of MLP (low-frequency, smooth components) and KAN’s sensitivity to high-frequency, nonlinear features, without alteration to overall network topologies. Domain decomposition in the modified formulation further reduces global optimization complexity by distributing frequency content and PDE variability over localized subnets.

Notable limitations are increased parameter count and wall-clock time (approximately double relative to single-branch approaches in standard implementations), the lack of an automated protocol for optimal $\xi$ (or $\alpha$) selection, and scaling challenges for very high-dimensional or large-scale PDEs. Prospective improvements include adaptive or learned branch fusion strategies and structural regularization via pruning or depth adjustment (Xu et al., 30 Mar 2025, Huang et al., 14 Nov 2025).

5. Overlapping Domain Decomposition and Adaptive Weighting

The extension of HPKM-PINN to include overlapping domain decomposition targets the curse of dimensionality, multiscale features, and high-frequency regimes often present in complex PDE systems (Huang et al., 14 Nov 2025). The algorithm consists of dividing the global domain $\Omega$ into overlapping subdomains $\Omega_i$, each equipped with a local HPKM architecture with an independently trained fusion weight. Transition and window functions $\omega_i(x)$ construct a partition of unity, ensuring global prediction smoothness and continuity. The convex weight between KAN/MLP local predictors is made learnable, with $S(\alpha^{(i)})$ (e.g., sigmoid mappings) steering representation capacity as necessitated by local frequency or regularity.

Empirical findings from benchmarks such as high-frequency 2D Helmholtz, 2D/5D Poisson, and nonlinear reaction–diffusion equations demonstrate that MHPKM-PINN consistently attains lower $L_2$ errors versus MLP-only and KAN-only baselines—with robustness to increasing frequency and dimensional complexity unattainable via other architectures at equivalent parameterization.

6. Context and Research Trajectory

HPKM-PINN and its domain-decomposed variant address key challenges that have limited the standard PINN paradigm, notably spectral bias toward low frequencies (MLP), and representational rigidity or parameter inefficiency (KAN). By adaptively blending branches—either via hand-tuned or trainable weights—and employing parallelism, hybridization, and domain partitioning, these methods furnish a systematic improvement in the modeling and computational resolution of challenging multiscale PDEs and oscillatory functional regression tasks. Open research directions include automated or regionally adaptive fusion mechanisms, scaling strategies for extreme dimensions, and further exploration of model pruning and depth-heterogeneity across subdomains (Xu et al., 30 Mar 2025, Huang et al., 14 Nov 2025).

7. Summary Table: Key HPKM-PINN Features and Comparisons

Feature	HPKM-PINN	PINN (MLP)	PIKAN (KAN)
Branch types	KAN + MLP (parallel)	MLP	KAN
Fusion weight	$\xi$ (fixed or learned)	---	---
Domain decomposition	Optional (modified model only)	No	No
Frequency handling	Adaptive (high/low)	Low freq. bias	High freq. bias
Typical error reduction	Up to $10^2\times$	---	---
Parameter cost	$\sim 2\times$ PINN/PIKAN	Baseline	Baseline
Robustness to noise	High	Moderate	Moderate-to-high

HPKM-PINN constitutes a versatile, modular enhancement within the PINN ecosystem, enabling tunable and efficient resolution of multi-frequency, high-dimensional, and noise-challenged PDEs by leveraging explicit feature fusion and, where needed, spatial domain partitioning (Xu et al., 30 Mar 2025, Huang et al., 14 Nov 2025).