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

2025-12-25

AI Summary

To address gradient flow stiffness and spectral bias in physics-informed neural networks (PINNs) when solving multiscale or high-frequency partial differential equations (PDEs), this work proposes the Physics-Informed Kolmogorov–Arnold Network (PI-KAN). Methodologically: (i) it introduces a novel KAN architecture integrated with learnable B-spline activations to enhance multiscale representation capacity; (ii) it designs an adaptive loss weighting scheme with dynamically decaying upper bounds to jointly regulate the optimization dynamics of physical constraints and data fidelity; and (iii) it achieves gradient flow stabilization without additional computational overhead. Benchmark evaluations on the Klein–Gordon, Burgers, and Helmholtz equations demonstrate significantly accelerated convergence, over one-order-of-magnitude improvement in accuracy, and markedly enhanced generalization. This work breaks fundamental PINN bottlenecks through dual innovations—network architecture and optimization mechanism.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Addresses stiffness and spectral bias in PINNs for multi-scale PDEs
• Enhances convergence and accuracy with adaptive weighting strategy
• Improves generalization in solving complex partial differential equations

Innovation

• Uses Kolmogorov-Arnold network with learnable B-splines
• Implements adaptive weighting with dynamic decay bound
• Accelerates convergence and boosts accuracy significantly