# BEKAN: Boundary condition-guaranteed evolutionary Kolmogorov-Arnold networks with radial basis functions for solving PDE problems

arXiv: https://arxiv.org/abs/2510.03576 · HTML: https://mlsanity.com/paper/2510.03576/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 42 references that are in the map.
- First posted on arXiv 3 Oct 2025.

## Builds on (most cited first, 22 of 22)

- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks](https://mlsanity.com/paper/2104.08426/) (2021)
- [Kolmogorov-Arnold Networks are Radial Basis Function Networks](https://mlsanity.com/paper/2405.06721/) (2024)
- [Multifidelity deep neural operators for efficient learning of partial differential equations with application to fast inverse design of nanoscale heat transport](https://mlsanity.com/paper/2204.06684/) (2022)
- [A survey on modern trainable activation functions](https://mlsanity.com/paper/2005.00817/) (2020)
- [On Interpretability of Artificial Neural Networks: A Survey](https://mlsanity.com/paper/2001.02522/) (2020)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness](https://mlsanity.com/paper/2305.17289/) (2023)
- [Evolutional Deep Neural Network](https://mlsanity.com/paper/2103.09959/) (2021)
- [Imposing Hard Constraints on Deep Networks: Promises and Limitations](https://mlsanity.com/paper/1706.02025/) (2017)
- [ExSpliNet: An interpretable and expressive spline-based neural network](https://mlsanity.com/paper/2205.01510/) (2022)
- [Linearly Constrained Gaussian Processes with Boundary Conditions](https://mlsanity.com/paper/2002.00818/) (2020)
- [Energy-Dissipative Evolutionary Deep Operator Neural Networks](https://mlsanity.com/paper/2306.06281/) (2023)
- [On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theorem](https://mlsanity.com/paper/2308.05509/) (2023)
- [BdryGP: a new Gaussian process model for incorporating boundary information](https://mlsanity.com/paper/1908.08868/) (2019)
- [A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs](https://mlsanity.com/paper/2210.03526/) (2022)
- [Know Your Boundaries: Constraining Gaussian Processes by Variational Harmonic Features](https://mlsanity.com/paper/1904.05207/) (2019)
- [Hard-constraining Neumann boundary conditions in physics-informed neural networks via Fourier feature embeddings](https://mlsanity.com/paper/2504.01093/) (2025)
- [PINN-FEM: A Hybrid Approach for Enforcing Dirichlet Boundary Conditions in Physics-Informed Neural Networks](https://mlsanity.com/paper/2501.07765/) (2025)

## Built on by (most cited first, 1 of 1)

- [Weak-Form Evolutionary Kolmogorov-Arnold Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2602.18515/) (2026)

## Abstract (arXiv)

Deep learning has gained attention for solving PDEs, but the black-box nature of neural networks hinders precise enforcement of boundary conditions. To address this, we propose a boundary condition-guaranteed evolutionary Kolmogorov-Arnold Network (KAN) with radial basis functions (BEKAN). In BEKAN, we propose three distinct and combinable approaches for incorporating Dirichlet, periodic, and Neumann boundary conditions into the network. For Dirichlet problem, we use smooth and global Gaussian RBFs to construct univariate basis functions for approximating the solution and to encode boundary information at the activation level of the network. To handle periodic problems, we employ a periodic layer constructed from a set of sinusoidal functions to enforce the boundary conditions exactly. For a Neumann problem, we devise a least-squares formulation to guide the parameter evolution toward satisfying the Neumann condition. By virtue of the boundary-embedded RBFs, the periodic layer, and the evolutionary framework, we can perform accurate PDE simulations while rigorously enforcing boundary conditions. For demonstration, we conducted extensive numerical experiments on Dirichlet, Neumann, periodic, and mixed boundary value problems. The results indicate that BEKAN outperforms both multilayer perceptron (MLP) and B-splines KAN in terms of accuracy. In conclusion, the proposed approach enhances the capability of KANs in solving PDE problems while satisfying boundary conditions, thereby facilitating advancements in scientific computing and engineering applications.