# 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: https://arxiv.org/abs/2409.20431 · HTML: https://mlsanity.com/paper/2409.20431/

## 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 (most cited first, 29 of 29)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning](https://mlsanity.com/paper/2008.13333/) (2020)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [An overview on deep learning-based approximation methods for partial differential equations](https://mlsanity.com/paper/2012.12348/) (2020)
- [Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks](https://mlsanity.com/paper/1905.01205/) (2019)
- [A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations](https://mlsanity.com/paper/1809.02362/) (2018)
- [Deep splitting method for parabolic PDEs](https://mlsanity.com/paper/1907.03452/) (2019)
- [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](https://mlsanity.com/paper/1809.03062/) (2018)
- [Convergence of the Deep BSDE Method for Coupled FBSDEs](https://mlsanity.com/paper/1811.01165/) (2018)
- [Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality](https://mlsanity.com/paper/2106.08900/) (2021)
- [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](https://mlsanity.com/paper/1809.07321/) (2018)
- [Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning](https://mlsanity.com/paper/2011.04602/) (2020)
- [Approximation error analysis of some deep backward schemes for nonlinear PDEs](https://mlsanity.com/paper/2006.01496/) (2020)
- [Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory](https://mlsanity.com/paper/2310.20360/) (2023)
- [Universal approximation property of Banach space-valued random feature models including random neural networks](https://mlsanity.com/paper/2312.08410/) (2023)
- [Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems](https://mlsanity.com/paper/1903.06652/) (2019)
- [Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs](https://mlsanity.com/paper/2405.05192/) (2024)
- [Deep learning based numerical approximation algorithms for stochastic partial differential equations](https://mlsanity.com/paper/2012.01194/) (2020)
- [Extensions of the Deep Galerkin Method](https://mlsanity.com/paper/1912.01455/) (2019)
- [A deep solver for BSDEs with jumps](https://mlsanity.com/paper/2211.04349/) (2022)
- [A neural network based policy iteration algorithm with global H2-superlinear convergence for stochastic games on domains](https://mlsanity.com/paper/1906.02304/) (2019)
- [Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for space-time solutions of semilinear partial differential equations](https://mlsanity.com/paper/2406.10876/) (2024)
- [Deep Neural Network Algorithms for Parabolic PIDEs and Applications in Insurance Mathematics](https://mlsanity.com/paper/2109.11403/) (2021)
- [A deep branching solver for fully nonlinear partial differential equations](https://mlsanity.com/paper/2203.03234/) (2022)
- [A deep learning approach to the probabilistic numerical solution of path-dependent partial differential equations](https://mlsanity.com/paper/2209.15010/) (2022)

## Built on by (most cited first, 3 of 3)

- [Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionality](https://mlsanity.com/paper/2506.22851/) (2025)
- [Walk-on-Spheres Monte Carlo and deep neural network approximations of elliptic PDEs with drift and killing](https://mlsanity.com/paper/2608.09494/) (2026)
- [Residual neural networks overcome the curse of dimensionality for semilinear heat equations](https://mlsanity.com/paper/2609.03626/) (2026)

## 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.