KP-PINNs: Kernel Packet Accelerated Physics Informed Neural Networks
This work addresses stability issues in PINNs for scientific computing applications, offering an incremental improvement over existing methods.
The paper tackles the problem of instability and inaccuracy in Physics Informed Neural Networks (PINNs) when using the L2 loss function for complex differential equations, proposing KP-PINNs with a new loss function based on RKHS norm and Kernel Packet acceleration, which achieves stable solutions across various equations.
Differential equations are involved in modeling many engineering problems. Many efforts have been devoted to solving differential equations. Due to the flexibility of neural networks, Physics Informed Neural Networks (PINNs) have recently been proposed to solve complex differential equations and have demonstrated superior performance in many applications. While the L2 loss function is usually a default choice in PINNs, it has been shown that the corresponding numerical solution is incorrect and unstable for some complex equations. In this work, we propose a new PINNs framework named Kernel Packet accelerated PINNs (KP-PINNs), which gives a new expression of the loss function using the reproducing kernel Hilbert space (RKHS) norm and uses the Kernel Packet (KP) method to accelerate the computation. Theoretical results show that KP-PINNs can be stable across various differential equations. Numerical experiments illustrate that KP-PINNs can solve differential equations effectively and efficiently. This framework provides a promising direction for improving the stability and accuracy of PINNs-based solvers in scientific computing.