🤖 AI Summary
Traditional Kolmogorov–Arnold networks (KANs) suffer from structural ambiguity, redundant variables, and poor trainability and scalability when deployed in physics-informed neural networks (PINNs). To address these limitations, we propose ActNet—a learnable deep model rigorously reconstructed based on the Kolmogorov superposition theorem (KST). Our key innovation lies in parameterizing the KST architecture with learnable activation functions and integrating physics-informed constraints within a PDE-driven unsupervised learning framework—enabling, for the first time, stable, trainable, and scalable deployment of KST in PINNs. Experiments demonstrate that ActNet significantly outperforms KANs across multiple PDE-solving benchmarks, matching the accuracy of state-of-the-art MLPs. Moreover, under label-free conditions, ActNet achieves high-fidelity implicit modeling of low-dimensional functions, exhibiting superior generalization and robustness. This work establishes a novel paradigm for function approximation and PDE solving in scientific computing.
📝 Abstract
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.