Data to 7 Oct 2026

Error bounds for approximations with deep ReLU neural networks in Ws,p norms

arXiv 1902.07896 ↗First posted 21 Feb 2019math.FA cs.LGcited by 95

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Abstract (arXiv)

We analyze approximation rates of deep ReLU neural networks for Sobolev-regular functions with respect to weaker Sobolev norms. First, we construct, based on a calculus of ReLU networks, artificial neural networks with ReLU activation functions that achieve certain approximation rates. Second, we establish lower bounds for the approximation by ReLU neural networks for classes of Sobolev-regular functions. Our results extend recent advances in the approximation theory of ReLU networks to the regime that is most relevant for applications in the numerical analysis of partial differential equations.

Key facts. Cited by 95 papers in the map (data through 7 Oct 2026). First posted on arXiv 21 Feb 2019.

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Citing papers per year: 2021: 18; 2022: 13; 2023: 19; 2024: 16; 2025: 16; 2026: 13.

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2023Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Datacited by 171
2021On the approximation of functions by tanh neural networkscited by 81
2024An Overview of Diffusion Models: Applications, Guided Generation, Statistical Rates and Optimizationcited by 72
2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
2022Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spacescited by 50
2021Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networkscited by 43
2021Universal Approximation Theorems for Differentiable Geometric Deep Learningcited by 41
2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
2023Deep Network Approximation: Beyond ReLU to Diverse Activation Functionscited by 33
2021Universal Approximation Under Constraints is Possible with Transformerscited by 32
2023Nearly Optimal VC-Dimension and Pseudo-Dimension Bounds for Deep Neural Network Derivativescited by 30

38 more

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2021Learning elliptic partial differential equations with randomized linear algebracited by 29
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2021A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equationscited by 28
2023Robust Multi-Agent Reinforcement Learning via Adversarial Regularization: Theoretical Foundation and Stable Algorithmscited by 25
2021Variational Physics Informed Neural Networks: the role of quadratures and test functionscited by 17
2022Neural Network Architecture Beyond Width and Depthcited by 17
2021Lyapunov-Net: A Deep Neural Network Architecture for Lyapunov Function Approximationcited by 16
2021Approximation Properties of Deep ReLU CNNscited by 15
2023Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirscited by 15
2021Error Estimates for the Deep Ritz Method with Boundary Penaltycited by 14
2024Deeper or Wider: A Perspective from Optimal Generalization Error with Sobolev Losscited by 14
2023On the Optimal Expressive Power of ReLU DNNs and Its Application in Approximation with Kolmogorov Superposition Theoremcited by 13
2024On the optimal approximation of Sobolev and Besov functions using deep ReLU neural networkscited by 13
2021ReLU Deep Neural Networks from the Hierarchical Basis Perspectivecited by 12
2021Solving PDEs on Unknown Manifolds with Machine Learningcited by 12
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2023Rates of Approximation by ReLU Shallow Neural Networkscited by 12
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2023Multilevel CNNs for Parametric PDEscited by 9
2023Non-asymptotic Approximation Error Bounds of Parameterized Quantum Circuitscited by 9
2025On the minimax optimality of Flow Matching through the connection to kernel density estimationcited by 9
2021Deep Neural Networks with ReLU-Sine-Exponential Activations Break Curse of Dimensionality in Approximation on H"older Classcited by 7
2022Approximation results for Gradient Descent trained Shallow Neural Networks in 1dcited by 7
2024Learning smooth functions in high dimensions: from sparse polynomials to deep neural networkscited by 7
2022Do ReLU Networks Have An Edge When Approximating Compactly-Supported Functions?cited by 6
2022Benefits of Overparameterized Convolutional Residual Networks: Function Approximation under Smoothness Constraintcited by 6
2023On Enhancing Expressive Power via Compositions of Single Fixed-Size ReLU Networkcited by 6
2023Neural Control of Parametric Solutions for High-dimensional Evolution PDEscited by 5
2023Residual Multi-Fidelity Neural Network Computingcited by 5
2024Quantitative Approximation for Neural Operators in Nonlinear Parabolic Equationscited by 5
2025Transformers Can Overcome the Curse of Dimensionality: A Theoretical Study from an Approximation Perspectivecited by 5
2025Deep Neural Networks with General Activations: Super-Convergence in Sobolev Normscited by 5
2025Distribution estimation via Flow Matching with Lipschitz guaranteescited by 5
2021Universal Regular Conditional Distributionscited by 4
2022Approximation Power of Deep Neural Networks: an explanatory mathematical surveycited by 4
2022Convergent autoencoder approximation of low bending and low distortion manifold embeddingscited by 4
2022SignReLU neural network and its approximation abilitycited by 4

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