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：本文探讨了柯尔莫哥洛夫叠加定理作为神经网络设计基础的替代表述。原始的KST表述虽然在数学上优雅，但由于其对内外函数结构的洞察有限以及引入了大量未知变量，带来了实际挑战。柯尔莫哥洛夫-阿诺德网络利用KST进行函数逼近，但与传统的多层感知机相比，其效果参差不齐，且受原始KST表述的实际限制，因此受到了审视。为解决这些问题，我们提出了ActNet，一种可扩展的深度学习模型，它建立在KST基础上，并克服了柯尔莫哥洛夫原始表述的许多缺点。我们在物理信息神经网络框架下评估ActNet，该框架非常适合利用KST在低维函数逼近方面的优势，特别是在模拟偏微分方程方面。在这一具有挑战性的场景中，模型必须在没有直接测量的情况下学习隐函数，ActNet在多个基准测试中始终优于KANs，并与当前最佳的基于MLP的方法具有竞争力。这些结果表明，ActNet为基于KST的深度学习应用，特别是在科学计算和PDE模拟任务中，提供了一个有前景的新方向。