(untitled) (Communications in Computational Physics, 2025)
This paper proposes a Kolmogorov high order deep neural network (K-HOrderDNN) for solving high-dimensional partial differential equations (PDEs), which improves the high order deep neural networks (HOrderDNNs). HOrderDNNs have been demonstrated to outperform conventional DNNs for high frequency problems by introducing a nonlinear transformation layer consisting of (p+1)^d basis functions. However, the number of basis functions grows exponentially with the dimension d, which results in the curse of dimensionality (CoD). Inspired by the Kolmogorov Superposition Theorem (KST), which expresses a multivariate function as superpositions of univariate functions and addition, K-HOrderDNN utilizes a HOrderDNN to efficiently approximate univariate inner functions instead of directly approximating the multivariate function, reducing the number of introduced basis functions to d(p+1). We theoretically demonstrate that CoD is mitigated when target functions belong to a dense subset of continuous multivariate functions. Extensive numerical experiments show that: for high-dimensional problems (d=10, 20, 50) where HOrderDNNs(p>1) are intractable, K-HOrderDNNs(p>1) exhibit remarkable performance. Specifically, when d=10, K-HOrderDNN(p=7) achieves an error of 4.40E−03, two orders of magnitude lower than that of HOrderDNN(p=1) (see Table 10); for high frequency problems, K-HOrderDNNs(p>1) can achieve higher accuracy with fewer parameters and faster convergence rates compared to HOrderDNNs (see Table 8).