A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients

深度ReLU神经网络在逼近半线性偏积分-微分方程时可克服维数灾难

Deep ReLU neural networks overcome the curse of dimensionality when approximating semilinear partial integro-differential equations

该研究针对带梯度无关Lipschitz非线性项的半线性PIDE，证明ReLU深度神经网络逼近其解时参数量仅随维度和精度倒数多项式增长，可克服维数灾难。

Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations

Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear e…

The BMO-discrepancy suffers from the curse of dimensionality

We show that the minimal discrepancy of a point set in the $d$-dimensional unit cube with respect to the BMO seminorm suffers from the curse of dimensionality.

Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

For a long time it is well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semi…

Deep neural network approximations for Monte Carlo algorithms

Recently, it has been proposed in the literature to employ deep neural networks (DNNs) together with stochastic gradient descent methods to approximate solutions of PDEs. There are also a few results in the literature which prove that DNNs can approximate solutions of certain PDEs without the curse of dimensionality in the sense that the number of real parameters used to describe the DNN grows at most polynomially both in the PDE dimension and the reciprocal of the prescribed approximation accuracy. One key argument in most of these results is, first, to use a Monte Carlo approximation scheme …

校正深度神经网络在逼近 McKean–Vlasov 随机微分方程解时克服维数灾难

Rectified deep neural networks overcome the curse of dimensionality when approximating solutions of McKean--Vlasov stochastic differential equations

该研究证明校正深度神经网络逼近 McKean–Vlasov SDE 解时参数数量仅随维数与精度倒数多项式增长，从而克服维数灾难。

Deep ReLU networks overcome the curse of dimensionality for bandlimited functions

We prove a theorem concerning the approximation of bandlimited multivariate functions by deep ReLU networks for which the curse of the dimensionality is overcome. Our theorem is based on a result by Maurey and on the ability of deep ReLU networks to approximate Chebyshev polynomials and analytic functions efficiently.

The Representation Power of Neural Networks: Breaking the Curse of Dimensionality

In this paper, we analyze the number of neurons and training parameters that a neural networks needs to approximate multivariate functions of bounded second mixed derivatives -- Korobov functions. We prove upper bounds on these quantities for shallow and deep neural networks, breaking the curse of dimensionality. Our bounds hold for general activation functions, including ReLU. We further prove that these bounds nearly match the minimal number of parameters any continuous function approximator needs to approximate Korobov functions, showing that neural networks are near-optimal function approx…

无维度灾难的复合函数深度神经网络逼近

Deep neural network approximation of composite functions without the curse of dimensionality

本文提出一类可由ReLU深度神经网络在无维度灾难条件下逼近的高维连续函数，其参数规模随输入维度和误差多项式增长，函数由乘积、最大值及并行Lipschitz连续函数任意复合而成。

修正深度神经网络在梯度依赖型半线性热方程数值逼近中克服维度灾难

Rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of gradient-dependent semilinear heat equations

本文首次严格证明深度神经网络在逼近一类具有梯度依赖非线性项的非线性偏微分方程时，能够克服维度灾难。

Approximation of Functionals by Neural Network without Curse of Dimensionality

In this paper, we establish a neural network to approximate functionals, which are maps from infinite dimensional spaces to finite dimensional spaces. The approximation error of the neural network is $O(1/\sqrt{m})$ where $m$ is the size of networks, which overcomes the curse of dimensionality. The key idea of the approximation is to define a Barron spectral space of functionals.

残差神经网络克服半线性热方程的维数灾难

Residual neural networks overcome the curse of dimensionality for semilinear heat equations

本文证明残差神经网络可克服半线性热方程的维数灾难，给出其参数数量的显式界，通过ResNet实现多级皮卡估计器的确定性完成近似任务。

Deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear partial differential equations

We prove that deep neural networks are capable of approximating solutions of semilinear Kolmogorov PDE in the case of gradient-independent, Lipschitz-continuous nonlinearities, while the required number of parameters in the networks grow at most polynomially in both dimension $d \in \mathbb{N}$ and prescribed reciprocal accuracy $\varepsilon$. Previously, this has only been proven in the case of semilinear heat equations.

全历史递归多层Picard逼近在随机控制问题的Hamilton-Jacobi-Bellman方程中遭遇维度灾难

Full history recursive multilevel Picard approximations suffer from the curse of dimensionality for the Hamilton-Jacobi-Bellman equation of a stochastic control problem

本文证明全历史递归多层Picard逼近在处理随机控制理论中具有局部Lipschitz非线性的Hamilton-Jacobi-Bellman方程时，会遭遇维度灾难。

Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations

One of the most challenging problems in applied mathematics is the approximate solution of nonlinear partial differential equations (PDEs) in high dimensions. Standard deterministic approximation methods like finite differences or finite elements suffer from the curse of dimensionality in the sense that the computational effort grows exponentially in the dimension. In this work we overcome this difficulty in the case of reaction-diffusion type PDEs with a locally Lipschitz continuous coervice nonlinearity (such as Allen-Cahn PDEs) by introducing and analyzing truncated variants of the recently…

多级Picard逼近在数值逼近含梯度依赖非线性项的一般半线性偏微分方程时可克服维数灾难

Multilevel Picard approximations overcome the curse of dimensionality in the numerical approximation of general semilinear PDEs with gradient-dependent nonlinearities

本文证明了Neufeld与Wu提出的用于求解含梯度依赖非线性项的一般半线性抛物型PDE的多级Picard（MLP）算法可克服维数灾难，其计算复杂度仅随维度和精度倒数呈多项式增长。

几何 $L_p$-差异的可处理性与维度诅咒

Tractability versus curse of dimensionality for geometric $L_p$-discrepancies

通过差异-积分对偶框架，研究了几何 $L_p$-差异的可处理性与维度诅咒，证明在张量积假设下非负线性规则的信息复杂度指数增长，导致维度诅咒。

多变量单调和凸函数的维数灾难

The Curse of Dimensionality for Monotone and Convex Functions of Many Variables

研究多变量单调和凸有界函数的积分与逼近问题，证明其存在维数灾难，即需要指数级（关于d）的函数值才能达到误差ε。

多层Picard逼近在 $L^p$ 意义下克服了具有梯度依赖非线性的半线性热方程的维数灾难

Multilevel Picard approximations overcome the curse of dimensionality when approximating semilinear heat equations with gradient-dependent nonlinearities in $L^p$-sense

本文证明在梯度依赖且Lipschitz连续的非线性条件下，多层Picard逼近能在 $L^p$ 意义下以多项式计算复杂度逼近半线性热方程的解，从而克服了维数灾难。

Ranks of Tensor Networks for Eigenspace Projections and the Curse of Dimensionality

The hierarchical (multi-linear) rank of an order-$d$ tensor is key in determining the cost of representing a tensor as a (tree) Tensor Network (TN). In general, it is known that, for a fixed accuracy, a tensor with random entries cannot be expected to be efficiently approximable without the curse of dimensionality, i.e., a complexity growing exponentially with $d$. In this work, we show that the ground state projection (GSP) of a class of unbounded Hamiltonians can be approximately represented as an operator of low effective dimensionality that is independent of the (high) dimension $d$ of the…

多级皮卡尔近似与具有ReLU、漏式ReLU和softplus激活函数的深度神经网络在以L^p意义近似半线性抛物型偏微分方程时克服维度灾难

Multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation overcome the curse of dimensionality when approximating semilinear parabolic partial differential equations in $L^p$-sense

研究证明，多级皮卡尔近似与具有ReLU、leaky ReLU和softplus激活函数的深度神经网络能以L^p意义近似半线性Kolmogorov PDE解，且计算复杂度和参数数量随维度和精度要求呈多项式增长。

在超立方体上恢复脊柱函数受到维度诅咒的困扰

The recovery of ridge functions on the hypercube suffers from the curse of dimensionality

本文研究了在超立方体上恢复未知脊柱函数时，即使允许随机算法，当用L∞范数衡量恢复误差时，仍然受到维度诅咒的限制，但若向量a的有限个分量显著大于其他分量，则问题可以弱可处理。

数值积分中光滑函数的维度诅咒 II

The Curse of Dimensionality for Numerical Integration of Smooth Functions II

本文研究了在最坏情况下，光滑函数的数值积分中维度诅咒的存在性，分析了不同光滑性条件下的必要和充分条件，探讨了无限可微函数的维度诅咒及其相关可处理性 条件。

Overcoming the curse of dimensionality for approximating Lyapunov functions with deep neural networks under a small-gain condition

We propose a deep neural network architecture for storing approximate Lyapunov functions of systems of ordinary differential equations. Under a small-gain condition on the system, the number of neurons needed for an approximation of a Lyapunov function with fixed accuracy grows only polynomially in the state dimension, i.e., the proposed approach is able to overcome the curse of dimensionality.

光滑函数数值积分的维数灾难

The Curse of Dimensionality for Numerical Integration of Smooth Functions

本文证明了对于C^r函数的多变量积分，存在维数灾难：达到误差ε所需的函数值个数大于c_r (1+γ)^d，其中c_r,γ>0，d为维数。

单适定节点下的多元插值——破除维数灾难

Multivariate Interpolation in Unisolvent Nodes -- Lifting the Curse of Dimensionality

该研究提出广义非张量型单适定节点概念，将Newton、Lagrange插值推广到任意维度，针对Trefethen函数达最优指数收敛速率，节点数次指数依赖维度以抵御维数灾难，还提出至多二次时间、线性内存的高效稳定插值算法。

Overcoming the curse of dimensionality in the numerical approximation of backward stochastic differential equations

Backward stochastic differential equations (BSDEs) belong nowadays to the most frequently studied equations in stochastic analysis and computational stochastics. BSDEs in applications are often nonlinear and high-dimensional. In nearly all cases such nonlinear high-dimensional BSDEs cannot be solved explicitly and it has been and still is a very active topic of research to design and analyze numerical approximation methods to approximatively solve nonlinear high-dimensional BSDEs. Although there are a large number of research articles in the scientific literature which analyze numerical approx…

突破维度诅咒的几种方法

Several Approaches to Break the Curse of Dimensionality

本文探讨了突破高维问题计算难度的几种方法，通过利用问题结构特性来克服维度诅咒，并介绍了信息基础复杂性理论。

克服数学金融中维度诅咒的算法（CoDeFi）

An algorithm (CoDeFi) for overcoming the curse of dimensionality in mathematical finance

本文提出CoDeFi算法，通过求解偏微分方程类如Kolmogorov型方程和Black-Scholes方程，解决高维问题中的维度诅咒问题，提供稳健可靠的数值方法。

极端 $L_p$ 差异、数值积分与维度灾难

Extreme $L_p$ discrepancy, numerical integration and the curse of dimensionality

本文通过将极端 $L_p$ 差异与一个对偶积分问题的最大误差等价，证明了对于所有 $p \in (1,\infty)$，极端 $L_p$ 差异遭受维度灾难。

具有非负权重的\(L_1\)差异受维度诅咒影响

The curse of dimensionality for geometric $L_1$-discrepancies with nonnegative weights

研究具有非负权重的\(L_1\)差异受维度诅咒问题，通过结合体积偏差概率测度变换与分数阶矩估计证明其倒数下界，此下界适用于等权重点集，论证利用权重非负性，不涉及任意有符号权重。

Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems

In this paper, we establish that for a wide class of controlled stochastic differential equations (SDEs) with stiff coefficients, the value functions of corresponding zero-sum games can be represented by a deep artificial neural network (DNN), whose complexity grows at most polynomially in both the dimension of the state equation and the reciprocal of the required accuracy. Such nonlinear stiff systems may arise, for example, from Galerkin approximations of controlled stochastic partial differential equations (SPDEs), or controlled PDEs with uncertain initial conditions and source terms. This …

Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations

The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined deep learning, revolutionized the area of artificial intelligence, machine learning, and data analysis. In particular, these methods have been applied to the numerical solution of high-dimensional partial differential equations with great success. Recent simulations indicate that deep learning-based algorithms are capable of overcoming the curse of dimensionality for the numerical solution of Kolmogorov equations, which are widely used i…

A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations

Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential equations (PDEs), which are widely used in finance, engineering, and natural sciences. In particular, simulations indicate that algorithms based on deep learning overcome the curse of dimensionality in the numerical approximation of solutions of semilinear PDEs. For certain linear PDEs this has also been proved mathematically. The key contribution of this article is to rigorously prove this for the first time for a …

柯尔莫哥洛夫叠加定理可在逼近高维函数时打破维数灾难

The Kolmogorov Superposition Theorem can Break the Curse of Dimensionality When Approximating High Dimensional Functions

该研究利用柯尔莫哥洛夫叠加定理，通过双隐层ReLU网络、LKB样条和矩阵交叉逼近技术，在逼近高维KL连续函数时打破维数灾难，为降维问题提供新方法。

Error bounds for deep ReLU networks using the Kolmogorov--Arnold superposition theorem

We prove a theorem concerning the approximation of multivariate functions by deep ReLU networks, for which the curse of the dimensionality is lessened. Our theorem is based on a constructive proof of the Kolmogorov--Arnold superposition theorem, and on a subset of multivariate continuous functions whose outer superposition functions can be efficiently approximated by deep ReLU networks.

高维偏微分方程的时空深度神经网络逼近

Space-time deep neural network approximations for high-dimensional partial differential equations

本文研究高维偏微分方程的求解，提出在时空区域上利用深度神经网络逼近Kolmogorov PDE的解，证明了该方法在PDE维度和精度倒数上至多呈多项式增长，从而克服了维度灾难。

Deep ReLU neural networks overcome the curse of dimensionality for partial integrodifferential equations

Deep neural networks (DNNs) with ReLU activation function are proved to be able to express viscosity solutions of linear partial integrodifferental equations (PIDEs) on state spaces of possibly high dimension $d$. Admissible PIDEs comprise Kolmogorov equations for high-dimensional diffusion, advection, and for pure jump Lévy processes. We prove for such PIDEs arising from a class of jump-diffusions on $\mathbb{R}^d$ that for any suitable measure $μ^d$ on $\mathbb{R}^d$ there exist constants $C,{\mathfrak{p}},{\mathfrak{q}}>0$ such that for every $\varepsilon \in (0,1]$ and for every $d\in \mat…

Overcoming the curse of dimensionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities

Partial differential equations (PDEs) are a fundamental tool in the modeling of many real world phenomena. In a number of such real world phenomena the PDEs under consideration contain gradient-dependent nonlinearities and are high-dimensional. Such high-dimensional nonlinear PDEs can in nearly all cases not be solved explicitly and it is one of the most challenging tasks in applied mathematics to solve high-dimensional nonlinear PDEs approximately. It is especially very challenging to design approximation algorithms for nonlinear PDEs for which one can rigorously prove that they do overcome t…

Solving the Kolmogorov PDE by means of deep learning

Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of financial derivatives. Kolmogorov PDEs and SDEs, respectively, can typically not be solved explicitly and it has been and still is an active topic of research to design and analyze numerical methods which are able to approximately solve Kolmogorov PDEs and SDEs, respectively. Nea…

非负权重的$L\log L$-Luxemburg偏差存在维数灾难

The $L\log L$-Luxemburg Discrepancy for Nonnegative Weights Suffers from the Curse of Dimensionality

本研究针对非负权重的$L\log L$-Luxemburg偏差，证明其信息复杂度存在维数灾难，相关结论也适用于等权重拟蒙特卡罗规则。

关于人工神经网络在Black-Scholes偏微分方程数值逼近中克服维数灾难的证明

A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations

本文首次从数学上证明，人工神经网络在数值逼近Black-Scholes偏微分方程时，所需参数数量在逼近精度倒数和PDE维度上至多多项式增长，从而克服了维数灾难。

克服在具有违约风险的金融衍生品近似定价中的维度诅咒

Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks

本文研究了如何通过全历史递归多级皮尔德算法克服在具有违约风险的金融衍生品近似定价中的维度诅咒，证明了该方法在计算效率上具有多项式可处理性。

Deep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality in Approximation on Hölder Class

In this paper, we construct neural networks with ReLU, sine and $2^x$ as activation functions. For general continuous $f$ defined on $[0,1]^d$ with continuity modulus $ω_f(\cdot)$, we construct ReLU-sine-$2^x$ networks that enjoy an approximation rate $\mathcal{O}(ω_f(\sqrt{d})\cdot2^{-M}+ω_{f}\left(\frac{\sqrt{d}}{N}\right))$, where $M,N\in \mathbb{N}^{+}$ denote the hyperparameters related to widths of the networks. As a consequence, we can construct ReLU-sine-$2^x$ network with the depth $5$ and width $\max\left\{\left\lceil2d^{3/2}\left(\frac{3μ}ε\right)^{1/α}\right\rceil,2\left\lceil\log_…

Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality

Artificial neural networks (ANNs) have become a very powerful tool in the approximation of high-dimensional functions. Especially, deep ANNs, consisting of a large number of hidden layers, have been very successfully used in a series of practical relevant computational problems involving high-dimensional input data ranging from classification tasks in supervised learning to optimal decision problems in reinforcement learning. There are also a number of mathematical results in the scientific literature which study the approximation capacities of ANNs in the context of high-dimensional target fu…

高维数值积分在一般域上的维度诅咒

The curse of dimensionality for numerical integration on general domains

本文研究了高维数值积分在最坏情况下的维度诅咒问题，证明了对于不同光滑函数类，当域为具有小直径的各向同性凸体且满足通用ψ₂估计时，数值积分的复杂度随维度呈指数增长。特别地，作者将2014年Hinrichs等人在J. Complexity上的结果扩展到整个范围2≤p≤∞，并提供了一种统一的方法。

有限 $p>1$ 的 $L_p$-偏差遭遇维度灾难

The $L_p$-discrepancy for finite $p>1$ suffers from the curse of dimensionality

本文证明了对于区间 $(1,\infty)$ 中的所有 $p$，$L_p$-偏差均遭遇维度灾难，仅 $p=1$ 的情况仍悬而未决，该结论源于对锚定 Sobolev 空间正求积公式最坏情况误差的更普遍证明。

高维偏微分方程的有限表达式逼近：无维数灾难

Finite Expression Approximation of High-Dimensional PDEs Without the Curse of Dimensionality

本文证明有限表达式符号表示类可克服高维PDE的维数灾难，通过多级皮卡德框架传播及构造确定性逼近，实现多项式复杂度保证，为有限表达式方法奠定理论基础。

求解线性柯尔莫哥洛夫偏微分方程的裁剪ReLU网络经验风险最小化误差分析

Error analysis for empirical risk minimization over clipped ReLU networks in solving linear Kolmogorov partial differential equations

本文针对线性柯尔莫哥洛夫偏微分方程，用截断技术将裁剪ReLU网络ERM的泛化结果推广到多项式增长初始函数，证明其打破维数灾难且适用于Black-Scholes方程与热方程，并建立了相应的逼近误差结果。

基于束法的在无维度灾难方法中的剪枝

Bundle-based pruning in the max-plus curse of dimensionality free method

本文提出利用束法高效解决非凸状态空间中最大加法基函数的剪枝问题，通过半正定规划实现量子门合成的精确松弛。

关于Loewner框架、Kolmogorov叠加定理与维度灾难

On the Loewner framework, the Kolmogorov superposition theorem, and the curse of dimensionality

本文扩展Loewner框架至任意参数数量的线性参数系统，通过变量解耦将复杂度从O(N^3)降至低于O(N^1.5)，驯服维度灾难，并类比Kolmogorov叠加定理，提出两种从数据直接构造多元有理函数实现的算法。

利用Gevrey正则性打破量子偏微分方程求解器中的维度灾难

Breaking the Curse of Dimensionality in Quantum PDE Solvers via Gevrey Regularity

该研究关联实值周期函数光滑度与量子态制备成本，利用Gevrey正则性提出高效量子PDE求解傅里叶方法，可转化为门计数，还构建多体量子模拟流水线获多项式级量子加速。

有限p下$L_p$偏差的维数灾难

The curse of dimensionality for the $L_p$-discrepancy with finite $p$

该研究解决了长期存在的开放问题，证明了所有形如$p=2\ell/(2\ell-1)$（$\ell\in\mathbb{N}$）的$(1,2]$区间内的p对应的$L_p$偏差均存在维数灾难，结论基于锚定Sobolev空间数值积分最坏误差的更一般性结果推导得出。

The necessity of depth for artificial neural networks to approximate certain classes of smooth and bounded functions without the curse of dimensionality

In this article we study high-dimensional approximation capacities of shallow and deep artificial neural networks (ANNs) with the rectified linear unit (ReLU) activation. In particular, it is a key contribution of this work to reveal that for all $a,b\in\mathbb{R}$ with $b-a\geq 7$ we have that the functions $[a,b]^d\ni x=(x_1,\dots,x_d)\mapsto\prod_{i=1}^d x_i\in\mathbb{R}$ for $d\in\mathbb{N}$ as well as the functions $[a,b]^d\ni x =(x_1,\dots, x_d)\mapsto\sin(\prod_{i=1}^d x_i) \in \mathbb{R} $ for $ d \in \mathbb{N} $ can neither be approximated without the curse of dimensionality by means…