TY - JOUR
T1 - A Kolmogorov High Order Deep Neural Network for High Frequency Partial Differential Equations in High Dimensions
AU - Zhang , Yaqin
AU - Li , Ke
AU - Chang , Zhipeng
AU - Liu , Xuejiao
AU - Huang , Yunqing
AU - Xiang , Xueshuang
JO - Communications in Computational Physics
VL - 1
SP - 181
EP - 222
PY - 2025
DA - 2025/07
SN - 38
DO - http://doi.org/10.4208/cicp.OA-2024-0095
UR - https://global-sci.org/intro/article_detail/cicp/24256.html
KW - Deep neural network, Kolmogorov Superposition Theorem, high-dimensional and high-frequency PDEs.
AB - <p style="text-align: justify;">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$(p&gt;1)$ are intractable, K-HOrderDNNs$(p&gt;1)$ 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$(p&gt;1)$ can achieve higher accuracy with fewer parameters and faster
convergence rates compared to HOrderDNNs (see Table 8).</p>