[2309.13722] Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Numerical Analysis arXiv:2309.13722 (math) [Submitted on 24 Sep 2023 ( v1 ), last revised 28 Apr 2026 (this version, v3)] Title: Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense Authors: Julia Ackermann , Arnulf Jentzen , Thomas Kruse , Benno Kuckuck , Joshua Lee Padgett View a PDF of the paper titled Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense, by Julia Ackermann and 4 other authors View PDF HTML (experimental) Abstract: Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed. The interest that these methods have generated in the literature is in large part due to simulations which appear to demonstrate that such DL methods have the capacity to overcome the curse of dimensionality (COD) for PDEs in the sense that the number of computational operations they require to achieve a certain approximation accuracy $\varepsilon\in(0,\infty)$ grows at most polynomially in the PDE dimension $d\in\mathbb N$ and the reciprocal of $\varepsilon$. While there is thus far no mathematical result that proves that one of such methods is indeed capable of overcoming the COD, there are now a number of rigorous results in the literature that show that deep neural networks (DNNs) have the expressive power to approximate PDE solutions without the COD in the sense that the number of parameters used to describe the approximating DNN grows at most polynomially in both the PDE dimension $d\in\mathbb N$ and the reciprocal of the approximation accuracy $\varepsilon>0$. Roughly speaking, in the literature it is has been proved for every $T>0$ that solutions $u_d\colon [0,T]\times\mathbb R^d\to \mathbb R$, $d\in\mathbb N$, of semilinear heat PDEs with Lipschitz continuous nonlinearities can be approximated by DNNs with ReLU activation at the terminal time in the $L^2$-sense without the COD provided that the initial value functions $\mathbb R^d\ni x\mapsto u_d(0,x)\in\mathbb R$, $d\in\mathbb N$, can be approximated by ReLU DNNs without the COD. It is the key contribution of this work to generalize this result by establishing this statement in the $L^p$-sense with $p\in(0,\infty)$ and by allowing the activation function to be more general covering the ReLU, the leaky ReLU, and the softplus activation functions as special cases. Comments: 53 pages Subjects: Numerical Analysis (math.NA) ; Machine Learning (cs.LG); Probability (math.PR) MSC classes: 65M15, 65C05, 68T07 (Primary) 60H35 (Secondary) Cite as: arXiv:2309.13722 [math.NA] (or arXiv:2309.13722v3 [math.NA] for this version) https://doi.org/10.48550/arXiv.2309.13722 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Benno Kuckuck [ view email ] [v1] Sun, 24 Sep 2023 18:58:18 UTC (45 KB) [v2] Tue, 24 Jun 2025 10:07:05 UTC (46 KB) [v3] Tue, 28 Apr 2026 23:18:52 UTC (46 KB) Full-text links: Access Paper: View a PDF of the paper titled Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense, by Julia Ackermann and 4 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.NA < prev | next > new | recent | 2023-09 Change to browse by: cs cs.LG cs.NA math math.PR References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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