基於 MLP 與 KAN 模型架構的物理資訊驅動神經網路於多維問題分析之研究

Abstract

物理資訊驅動神經網路（Physics-Informed Neural Networks, PINNs）已成為求解偏微分方程（Partial Differential Equations, PDEs）的有力工具，具有結合物理法則與資料驅動模型的彈性。然而，傳統多層感知器（Multilayer Perceptron, MLP）在神經元上採用固定的激勵函數；相較之下，Kolmogorov–Arnold 網路（KAN）則在邊上引入可學習的激勵函數，以探索不同網路架構的可能性。本論文系統性地比較了三種代表性架構——MLP、KAN 與 Chebyshev 多項式為基礎的 KAN（Chebyshev-KAN），並於二維與三維的 Poisson 方程與 Navier–Cauchy 方程上進行實驗，針對準確度、收斂性、參數效率與計算成本進行分析與比較。

實驗結果顯示，MLP 在低複雜度問題中能提供穩定的準確度與泛化能力，但在高複雜度情境下則需透過增加網路寬度或深度來調整架構。相較之下，KAN 模型透過調整內部網格架構展現了更強的近似能力與收斂性，但同時伴隨較高的訓練時間與 GPU 記憶體需求。Chebyshev-KAN 則展現了兼具實用性與效率的特點，不僅保留了 KAN 的準確度優勢，還大幅減少了參數數量與計算成本，因而在效率與準確度之間達成有效平衡。

Physics-Informed Neural Networks (PINNs) have become a promising framework for solving partial differential equations (PDEs), offering flexibility in integrating physical laws with data-driven models. However, MLPs have fixed activation functions on neurons, and KANs provide learnable activation functions on edges which aim to explore the opportunities for different architectures. This thesis systematically investigates the performance of three representative architectures—Multilayer Perceptron (MLP), Kolmogorov–Arnold Network (KAN), and Chebyshev Polynomial-Based KAN (Chebyshev-KAN)—across two- and three-dimensional Poisson and Navier–Cauchy equations and makes comparison of accuracy, convergence, parameter efficiency, and computational cost.

The experimental results reveal that MLPs provide stable accuracy and generalization; however, under high-complexity situations they require adjustments toward wider or deeper architectures. KAN models, by contrast, exhibit strong approximation capability and improved convergence when modifing the internal grid architecture of KAN models, yet incur higher training time and GPU memory usage. Chebyshev-KANs emerge as a practical method, which preserves accuracy advantages of KANs while significantly reducing parameter counts and computational costs, thereby achieving an effective balance between efficiency and accuracy.