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).