This paper explores alternative formulations of the Kolmogorov Superposition Theorem (KST) as a foundation for neural network design. The original KST formulation, while mathematically elegant, presents practical challenges due to its limited insight into the structure of inner and outer functions and the large number of unknown variables it introduces. Kolmogorov-Arnold Networks (KANs) leverage KST for function approximation, but they have faced scrutiny due to mixed results compared to traditional multilayer perceptrons (MLPs) and practical limitations imposed by the original KST formulation. To address these issues, we introduce ActNet, a scalable deep learning model that builds on the KST and overcomes many of the drawbacks of Kolmogorov's original formulation. We evaluate ActNet in the context of Physics-Informed Neural Networks (PINNs), a framework well-suited for leveraging KST's strengths in low-dimensional function approximation, particularly for simulating partial differential equations (PDEs). In this challenging setting, where models must learn latent functions without direct measurements, ActNet consistently outperforms KANs across multiple benchmarks and is competitive against the current best MLP-based approaches. These results present ActNet as a promising new direction for KST-based deep learning applications, particularly in scientific computing and PDE simulation tasks.

翻译：本文探討科爾莫戈羅夫疊加定理（KST）的替代表述形式，作為神經網路設計的基礎。原始KST表述雖然在數學上優雅，但由於其對內外函數結構的洞察有限，且引入了大量未知變量，因此在實際應用中面臨挑戰。科爾莫戈羅夫-阿諾德網路（KANs）利用KST進行函數逼近，但由於其與傳統多層感知器（MLPs）相比效果參差不齊，且受原始KST表述的實際限制，一直備受質疑。為解決這些問題，我們引入了ActNet，這是一種可擴展的深度學習模型，它建立在KST基礎之上，並克服了科爾莫戈羅夫原始表述的許多缺點。我們在物理資訊神經網路（PINNs）的背景下評估ActNet，該框架特別適合利用KST在低維函數逼近方面的優勢，尤其是用於模擬偏微分方程（PDEs）。在這種具有挑戰性的環境中，模型必須在無直接測量值的情況下學習潛在函數，ActNet在多個基準測試中持續優於KANs，並且與當前最先進的基於MLP的方法相比具有競爭力。這些結果表明，ActNet為基於KST的深度學習應用，特別是在科學計算和PDE模擬任務中，開闢了一個有前景的新方向。