mlsanityData to 14 Oct 2026

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 Lp-sense

arXiv 2409.20431 ↗First posted 30 Sep 2024math.NA cs.LG cs.NAcited by 3

Not on the map. Why?

Abstract (arXiv)

We prove that multilevel Picard approximations and deep neural networks with ReLU, leaky ReLU, and softplus activation are capable of approximating solutions of semilinear Kolmogorov PDEs in L^mathfrakp-sense, mathfrakp∈ [2,∞), in the case of gradient-independent, Lipschitz-continuous nonlinearities, while the computational effort of the multilevel Picard approximations and the required number of parameters in the neural networks grow at most polynomially in both dimension d∈ mathbbN and reciprocal of the prescribed accuracy ε.

Key facts. Cited by 3 papers in the map (data through 14 Oct 2026). Builds on 29 of its 83 references that are in the map. First posted on arXiv 30 Sep 2024.

Builds on

29 of 83 references.

References · 29

  1. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  2. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
  3. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 583
  4. 2017Solving high-dimensional partial differential equations using deep learningcited by 556
  5. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 191
  6. 2020Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learningcited by 77
  7. 2017Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equationscited by 65
  8. 2020An overview on deep learning-based approximation methods for partial differential equationscited by 65
  9. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 64
  10. 2018A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equationscited by 55
  11. 2019Deep splitting method for parabolic PDEscited by 53
  12. 2018Analysis 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 Equationscited by 45
17 more
  1. 2018Convergence of the Deep BSDE Method for Coupled FBSDEscited by 43
  2. 2021Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionalitycited by 34
  3. 2018A 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 coefficientscited by 31
  4. 2020Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learningcited by 22
  5. 2020Approximation error analysis of some deep backward schemes for nonlinear PDEscited by 21
  6. 2023Mathematical Introduction to Deep Learning: Methods, Implementations, and Theorycited by 20
  7. 2023Universal approximation property of Banach space-valued random feature models including random neural networkscited by 13
  8. 2019Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systemscited by 12
  9. 2024Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEscited by 12
  10. 2020Deep learning based numerical approximation algorithms for stochastic partial differential equationscited by 11
  11. 2019Extensions of the Deep Galerkin Methodcited by 10
  12. 2022A deep solver for BSDEs with jumpscited by 9
  13. 2019A neural network based policy iteration algorithm with global H2-superlinear convergence for stochastic games on domainscited by 7
  14. 2024Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for space-time solutions of semilinear partial differential equationscited by 6
  15. 2021Deep Neural Network Algorithms for Parabolic PIDEs and Applications in Insurance Mathematicscited by 5
  16. 2022A deep branching solver for fully nonlinear partial differential equationscited by 3
  17. 2022A deep learning approach to the probabilistic numerical solution of path-dependent partial differential equationscited by 2

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 2.

3 papers.

  1. 2025Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionalitycited by 1
  2. 2026Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killingcited by 0
  3. 2026Residual neural networks overcome the curse of dimensionality for semilinear heat equationscited by 0