A Kolmogorov High Order Deep Neural Network for High Frequency Partial Differential Equations in High Dimensions (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), whichimproves the high order deep neural networks (HOrderDNNs). HOrderDNNs havebeen demonstrated to outperform conventional DNNs for high frequency problemsby 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 superpositionsof univariate functions and addition, K-HOrderDNN utilizes a HOrderDNN to efficiently approximate univariate inner functions instead of directly approximating themultivariate function, reducing the number of introduced basis functions to $d(p+1).$ We theoretically demonstrate that CoD is mitigated when target functions belong to adense subset of continuous multivariate functions. Extensive numerical experimentsshow 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 fasterconvergence rates compared to HOrderDNNs (see Table 8).