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arxiv: 2309.13722 · v3 · submitted 2023-09-24 · 🧮 math.NA · cs.LG· cs.NA· math.PR

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

Julia Ackermann , Arnulf Jentzen , Thomas Kruse , Benno Kuckuck , Joshua Lee Padgett This is my paper

Pith reviewed 2026-05-24 06:50 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NAmath.PR
keywords deep neural networkscurse of dimensionalitysemilinear heat equationsL^p approximationReLU activationKolmogorov PDEsLipschitz nonlinearities
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The pith

Solutions to high-dimensional semilinear heat PDEs with Lipschitz nonlinearities can be approximated in the L^p sense by deep neural networks with ReLU, leaky ReLU or softplus activations without the curse of dimensionality, provided the初始值

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper proves that if initial value functions for a family of semilinear heat equations can be approximated by deep neural networks with ReLU, leaky ReLU or softplus activations without the curse of dimensionality, then the same holds for the solutions at any positive terminal time in the L^p norm for any p between 0 and infinity. The result covers Kolmogorov PDEs whose nonlinear terms are Lipschitz continuous. A reader should care because it supplies a rigorous guarantee that the number of network parameters needed to reach a given accuracy grows only polynomially with dimension and with the reciprocal of the error tolerance, extending earlier L^2 statements to a wider range of error measures and activation functions.

Core claim

The paper establishes that for every T > 0 the solutions u_d : [0,T] × R^d → R of semilinear heat PDEs with Lipschitz continuous nonlinearities can be approximated at time T in the L^p sense, p ∈ (0,∞), by DNNs with ReLU, leaky ReLU or softplus activations without the curse of dimensionality whenever the initial functions x ↦ u_d(0,x) admit such approximations without the curse of dimensionality.

What carries the argument

Transfer of non-curse-of-dimensionality approximability from initial data to terminal-time solutions via the PDE evolution operator, for ReLU, leaky ReLU and softplus activations.

Load-bearing premise

The initial value functions themselves can be approximated without the curse of dimensionality by networks with the given activations.

What would settle it

Finding a sequence of initial functions approximable without the COD by the networks, but whose evolved solutions at time T require exponentially many parameters in d for the same accuracy in L^p.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that, for any fixed T>0 and p in (0,infty), solutions u_d of semilinear heat equations with Lipschitz nonlinearities on [0,T] x R^d can be approximated at time T in the L^p norm by DNNs using ReLU, leaky ReLU or softplus activations, with the number of parameters growing at most polynomially in d and 1/epsilon, provided the initial data functions admit such DNN approximations without the curse of dimensionality. This extends earlier results that were restricted to the L^2 norm and the ReLU activation.

Significance. If the proofs hold, the result supplies a clean technical extension of known approximation-theoretic guarantees for DNNs applied to high-dimensional Kolmogorov PDEs. Broadening the admissible activations and the range of p strengthens the theoretical case that DNNs can overcome the curse of dimensionality for this class of PDEs, conditional on the initial-data hypothesis that is already standard in the literature.

minor comments (2)
  1. The dependence of the constants on p and on the Lipschitz constant of the nonlinearity should be stated explicitly in the main theorem statement to make the polynomial-in-d claim fully transparent.
  2. Notation for the DNN parameter count N(d,epsilon) is used before it is formally defined; a forward reference or early definition would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive summary, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

Conditional result on external initial-data assumption; minor self-citations not load-bearing

full rationale

The paper establishes a conditional statement that terminal-time L^p approximability by DNNs (ReLU/leaky ReLU/softplus) follows from the assumption that initial-value functions can be approximated without the COD. This generalizes prior L^2/ReLU results via technical extension of error-propagation arguments rather than any internal reduction of the target quantity to a fitted parameter or self-defined input. The weakest assumption is explicitly external and not derived within the paper. Self-citations to earlier works appear but are not load-bearing for the new generalization, satisfying the criteria for at most minor circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on standard properties of the listed activation functions and on the external assumption that initial data admit DNN approximations without COD; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption ReLU, leaky ReLU and softplus satisfy the approximation-theoretic properties used in the prior L^2 result
    Invoked to extend the earlier theorem to the new activations.
  • domain assumption Solutions of the semilinear heat PDE with Lipschitz nonlinearity admit the regularity needed for the approximation argument
    Standard background fact for Kolmogorov PDEs.

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