DBAW-PIKAN: Dynamic Balance Adaptive Weight Kolmogorov-Arnold Neural Network for Solving Partial Differential Equations

TL;DR

This work tackles the limitations of physics-informed neural networks (PINNs) in multi-scale and high-frequency PDEs by jointly addressing gradient flow instabilities and spectral bias. It introduces DBAW-PIKAN, which couples the expressive Physics-Informed Kolmogorov-Arnold Network (PIKAN) with a Dynamic Balancing Adaptive Weighting (DBAW) scheme that uses a decaying upper bound $\gamma(t)$ to regulate loss-term contributions, with $\lambda_j = 1/(\sigma_j^2 + 1/\gamma(t) + \epsilon)$. The approach yields faster convergence and higher accuracy across Klein-Gordon, Burgers, and Helmholtz tests, outperforming PINN, DBAW-PINN, and PIKAN, and demonstrating a synergistic effect between high-expressivity architectures and robust optimization. This framework offers a principled path toward stable, accurate PDE solvers that can handle complex, multi-constraint problems in scientific computing. The results highlight the potential of combining edge-based spline activations with dynamic, uncertainty-informed loss weighting for scalable, high-precision PDE solvers.

Abstract

Physics-informed neural networks (PINNs) have led to significant advancements in scientific computing by integrating fundamental physical principles with advanced data-driven techniques. However, when dealing with problems characterized by multi-scale or high-frequency features, PINNs encounter persistent and severe challenges related to stiffness in gradient flow and spectral bias, which significantly limit their predictive capabilities. To address these issues, this paper proposes a Dynamic Balancing Adaptive Weighting Physics-Informed Kolmogorov-Arnold Network (DBAW-PIKAN), designed to mitigate such gradient-related failure modes and overcome the bottlenecks in function representation. The core of DBAW-PIKAN combines the Kolmogorov-Arnold network architecture, based on learnable B-splines, with an adaptive weighting strategy that incorporates a dynamic decay upper bound. Compared to baseline models, the proposed method accelerates the convergence process and improves solution accuracy by at least an order of magnitude without introducing additional computational complexity. A series of numerical benchmarks, including the Klein-Gordon, Burgers, and Helmholtz equations, demonstrate the significant advantages of DBAW-PIKAN in enhancing both accuracy and generalization performance.

Figures (8)

• Figure 1: PINN Framework Diagram
• Figure 2: DBAW-PIKAN Framework Diagram
• Figure 3: Numerical solution comparisons of the Klein-Gordon equation among four different methods. Each row displays (from left to right) the exact solution, the predicted solution, and the point-wise absolute error distribution for each respective framework.
• Figure 4: Parameter evolution during the training process for the Klein-Gordon equation: (a) Evolution of adaptive weights ($\lambda_r$, $\lambda_{ic}$, $\lambda_{bc}$) over training epochs; (b) Evolution of the dynamic upper bound parameter $\gamma(t)$ over training epochs; (c) Evolution of individual loss terms over training epochs. The y-axis is on a logarithmic scale, and the vertical dashed line indicates the epoch at which the best performance was achieved.
• Figure 5: Parameter evolution during the training process for the Burgers equation: (a) trajectories of adaptive weights ($\lambda_r$, $\lambda_{ic}$, $\lambda_{bc}$) over training epochs; (b) evolution of the dynamic upper bound parameter $\gamma(t)$ over training epochs; and (c) evolution of individual loss components over training epochs.