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A Theoretical Analysis of Deep Neural Networks and Parametric PDEs

arXiv 1904.00377 ↗First posted 31 Mar 2019math.NA cs.LG cs.NAcited by 74

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Abstract (arXiv)

We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent low-dimensionality of the solution manifold to obtain approximation rates which are significantly superior to those provided by classical neural network approximation results. Concretely, we use the existence of a small reduced basis to construct, for a large variety of parametric partial differential equations, neural networks that yield approximations of the parametric solution maps in such a way that the sizes of these networks essentially only depend on the size of the reduced basis.

Key facts. Cited by 74 papers in the map (data through 7 Oct 2026). First posted on arXiv 31 Mar 2019.

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Citing papers per year: 2021: 21; 2022: 13; 2023: 10; 2024: 15; 2025: 11; 2026: 4.

74 papers.

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  1. 2024Operator Learning: Algorithms and Analysiscited by 84
  2. 2021On the approximation of functions by tanh neural networkscited by 81
  3. 2021POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decompositioncited by 79
  4. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  5. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  6. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
  7. 2021Exponential Convergence of Deep Operator Networks for Elliptic Partial Differential Equationscited by 44
  8. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  9. 2021Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionalitycited by 34
  10. 2022Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problemscited by 34
  11. 2021Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spacescited by 30
  12. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
38 more
  1. 2021A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equationscited by 28
  2. 2021Two-layer neural networks with values in a Banach spacecited by 26
  3. 2021Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniquescited by 23
  4. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  5. 2021Parametric Complexity Bounds for Approximating PDEs with Neural Networkscited by 18
  6. 2021Variational Physics Informed Neural Networks: the role of quadratures and test functionscited by 17
  7. 2021Learning Density Distribution of Reachable States for Autonomous Systemscited by 11
  8. 2021An artificial neural network approach to bifurcating phenomena in computational fluid dynamicscited by 11
  9. 2023CS4ML: A general framework for active learning with arbitrary data based on Christoffel functionscited by 11
  10. 2024Agnostic Active Learning of Single Index Models with Linear Sample Complexitycited by 11
  11. 2021Connections between Numerical Algorithms for PDEs and Neural Networkscited by 10
  12. 2022Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order modelscited by 10
  13. 2023Multilevel CNNs for Parametric PDEscited by 9
  14. 2024A practical existence theorem for reduced order models based on convolutional autoencoderscited by 9
  15. 2024Operator Learning of Lipschitz Operators: An Information-Theoretic Perspectivecited by 9
  16. 2022Approximation results for Gradient Descent trained Shallow Neural Networks in 1dcited by 7
  17. 2022CAS4DL: Christoffel Adaptive Sampling for function approximation via Deep Learningcited by 6
  18. 2023A Deep Learning algorithm to accelerate Algebraic Multigrid methods in Finite Element solvers of 3D elliptic PDEscited by 6
  19. 2022Solving parametric partial differential equations with deep rectified quadratic unit neural networkscited by 5
  20. 2023Residual Multi-Fidelity Neural Network Computingcited by 5
  21. 2025Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programscited by 5
  22. 2025Variationally correct operator learning: Reduced basis neural operator with a posteriori error estimationcited by 5
  23. 2021Deep neural network surrogates for non-smooth quantities of interest in shape uncertainty quantificationcited by 4
  24. 2024FBSJNN: A Theoretically Interpretable and Efficiently Deep Learning method for Solving Partial Integro-Differential Equationscited by 4
  25. 2025Time to Spike? Understanding the Representational Power of Spiking Neural Networks in Discrete Timecited by 4
  26. 2021Designing Rotationally Invariant Neural Networks from PDEs and Variational Methodscited by 3
  27. 2022From Monte Carlo to neural networks approximations of boundary value problemscited by 3
  28. 2024A Low Rank Neural Representation of Entropy Solutionscited by 3
  29. 2021Approximation properties of Residual Neural Networks for Kolmogorov PDEscited by 2
  30. 2021Error Bounds for a Matrix-Vector Product Approximation with Deep ReLU Neural Networkscited by 2
  31. 2022Error analysis for deep neural network approximations of parametric hyperbolic conservation lawscited by 2
  32. 2022Limitations of neural network training due to numerical instability of backpropagationcited by 2
  33. 2023PINNs error estimates for nonlinear equations in ℝ-smooth Banach spacescited by 2
  34. 2023Approximation Results for Gradient Descent trained Neural Networkscited by 2
  35. 2023Approximating Langevin Monte Carlo with ResNet-like Neural Network architecturescited by 2
  36. 2024Adaptive Multilevel Neural Networks for Parametric PDEs with Error Estimationcited by 2
  37. 2025AsyReC: A Multimodal Graph-based Framework for Spatio-Temporal Asymmetric Dyadic Relationship Classificationcited by 2
  38. 2025Noisy PDE Training Requires Bigger PINNscited by 2

50 most cited of 74 All 74 (JSON)