SPIKANs: Separable Physics-Informed Kolmogorov-Arnold Networks

TL;DR

The paper addresses the high computational burden of physics-informed neural networks (PINNs) and Kolmogorov-Arnold networks (KANs) for high-dimensional PDEs. It introduces Separable Physics-Informed Kolmogorov-Arnold Networks (SPIKANs), which approximate the solution with a separable representation $u(oldsymbol{x},t) \approx \sum_{j=1}^{r} \prod_{i=1}^{n} f_j^{(\theta_i)}(x_i)$, training $d$ univariate networks of rank $r$ instead of a single multivariate network, thereby reducing time and memory. Across four benchmarks (2D Helmholtz, 2D cavity, 1D+1 Allen-Cahn, 2D+1 Klein-Gordon), SPIKANs achieved speedups in the range $O(10)$–$O(100)$ while delivering equal or improved accuracy compared with PIKANs. This work demonstrates scalable physics-informed learning for higher-dimensional PDEs and points to future extensions such as adaptive grids, multi-fidelity training, and separable DeepONets to further enhance performance.

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a promising method for solving partial differential equations (PDEs) in scientific computing. While PINNs typically use multilayer perceptrons (MLPs) as their underlying architecture, recent advancements have explored alternative neural network structures. One such innovation is the Kolmogorov-Arnold Network (KAN), which has demonstrated benefits over traditional MLPs, including faster neural scaling and better interpretability. The application of KANs to physics-informed learning has led to the development of Physics-Informed KANs (PIKANs), enabling the use of KANs to solve PDEs. However, despite their advantages, KANs often suffer from slower training speeds, particularly in higher-dimensional problems where the number of collocation points grows exponentially with the dimensionality of the system. To address this challenge, we introduce Separable Physics-Informed Kolmogorov-Arnold Networks (SPIKANs). This novel architecture applies the principle of separation of variables to PIKANs, decomposing the problem such that each dimension is handled by an individual KAN. This approach drastically reduces the computational complexity of training without sacrificing accuracy, facilitating their application to higher-dimensional PDEs. Through a series of benchmark problems, we demonstrate the effectiveness of SPIKANs, showcasing their superior scalability and performance compared to PIKANs and highlighting their potential for solving complex, high-dimensional PDEs in scientific computing.

Figures (9)

• Figure 1: Schematic of the SPIKAN architecture. We decompose the $n$-dimensional PDE into $n$ individual KANs. In contrast with PIKANs, which take as input points given by tuples $(x_1, ..., x_n)$, in SPIKANs, each network receives only one coordinate $x_i$ at a time. Each network produces $r$ outputs, which are then combined via an outer product and summation on the latent dimension (denoted by the $\odot$ operator) to approximate the field $\hat{u}(x_1,..., x_n)$. We use forward-mode automatic differentiation to evaluate the derivatives of $\hat{u}$ needed to satisfy the differential operator $\mathcal{D}$ and any other derivative needed to construct the losses. The weights of the networks are then adjusted via gradient descent.
• Figure 2: Contour plots and absolute error between the predicted solution $u$ obtained with PIKANs, SPIKANs and the analytical solution of the 2D Helmholtz boundary value problem. A description of the cases is shown in Table \ref{['tab:2dhelmholtz']}.
• Figure 3: Comparison of horizontal and vertical velocity profiles obtained with PIKAN ($[2,5,5,3], n_\text{cp}=50^2)$) and SPIKANs ($2 \times [1,9,9,3*5], n_\text{cp}=50^2, 100^2$) with the reference ghia1982high for $Re = 100$.
• Figure 4: Comparison of the isocontours of $||\mathbf{u}|| = \sqrt{u^2 + v^2}$ and streamlines, obtained with finite volume method, PIKAN ($[2,9,9,3], n_\text{cp}=50^2)$), SPIKAN (a) ($2 \times [1,5,5,3*5], n_\text{cp}=50^2$) and SPIKAN (b) ($2 \times [1,5,5,3*5], n_\text{cp}=100^2$).
• Figure 5: Comparison of absolute difference of predicted $u, v$ and $p$ fields obtained from PIKAN ($[2,9,9,3], n_\text{cp}=50^2$) and SPIKAN ($2 \times [1,5,5,3*5], n_\text{cp}=50^2$) versus the results obtained with FVM for $Re = 100$.