PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs

Imagine you're working on a complex engineering problem, like predicting how materials behave under stress or identifying parameters in a physical model. Traditionally, people use multilayer perceptrons (MLPs) to solve these partial differential equations (PDEs), but they often struggle with accuracy and interpretability due to their dense structure and spectral bias. This leads to issues like needing high-order derivatives for strong-form residuals or facing trivial solutions in energy formulations, which can be frustrating when trying to model real-world phenomena accurately. This is what's called the limitations of MLPs in physics-informed learning.

To address these challenges, the authors propose a new approach called PG-KINN, which utilizes Kolmogorov Arnold Networks (KANs) in a Petrov-Galerkin framework. The idea is to combine the strengths of KANs, which have learnable spline activations that align well with classical discretization methods, with a more flexible test space that can handle a variety of PDEs, including nonlinear and inverse problems. By integrating by parts, they reduce the differentiation order needed while maintaining the ability to tackle complex scenarios. The results show that PG-KINN consistently outperforms traditional MLPs and other KAN-based methods across various benchmarks, indicating a promising direction for AI in computational mechanics.

The introduction of Petrov-Galerkin formulations with KANs represents a significant methodological advancement in physics-informed learning.

The results are validated against a comprehensive suite of benchmarks, demonstrating consistent performance improvements.