Pith. sign in

REVIEW 2 minor 1 cited by

Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense

T0 review · 0 major / 2 minor · reviewed 2026-05-24 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗

Pith's one-line read Solutions to high-dimensional semilinear heat PDEs with Lipschitz nonlinearities can be approximated in the L^p sense by deep neural networks with ReLU, leaky ReLU or softplus activations without the curse of dimensionality, provided the初始值

desk verdict This paper extends the prior L2-ReLU result on DNN approximation of semilinear heat PDEs to Lp norms and leaky ReLU/softplus activations under the unchanged initial-data assumption. read the letter →

arxiv 2309.13722 v3 submitted 2023-09-24 math.NA cs.LGcs.NAmath.PR

Julia Ackermann , Arnulf Jentzen , Thomas Kruse , Benno Kuckuck , Joshua Lee Padgett This is my paper · ORCID
classification math.NAcs.LGcs.NAmath.PR
keywords deepneuralnetworkscurseofdimensionalitysemilinearheatequationsL^papproximationReLUactivationKolmogorovPDEsLipschitznonlinearities
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proves that if initial value functions for a family of semilinear heat equations can be approximated by deep neural networks with ReLU, leaky ReLU or softplus activations without the curse of dimensionality, then the same holds for the solutions at any positive terminal time in the L^p norm for any p between 0 and infinity. The result covers Kolmogorov PDEs whose nonlinear terms are Lipschitz continuous. A reader should care because it supplies a rigorous guarantee that the number of network parameters needed to reach a given accuracy grows only polynomially with dimension and with the reciprocal of the error tolerance, extending earlier L^2 statements to a wider range of error measures and activation functions.

What carries the argument

Transfer of non-curse-of-dimensionality approximability from initial data to terminal-time solutions via the PDE evolution operator, for ReLU, leaky ReLU and softplus activations.

What would settle it

Finding a sequence of initial functions approximable without the COD by the networks, but whose evolved solutions at time T require exponentially many parameters in d for the same accuracy in L^p.

Watch

Extended reading notes

Core claim

The paper establishes that for every T > 0 the solutions u_d : [0,T] × R^d → R of semilinear heat PDEs with Lipschitz continuous nonlinearities can be approximated at time T in the L^p sense, p ∈ (0,∞), by DNNs with ReLU, leaky ReLU or softplus activations without the curse of dimensionality whenever the initial functions x ↦ u_d(0,x) admit such approximations without the curse of dimensionality.

Load-bearing premise

The initial value functions themselves can be approximated without the curse of dimensionality by networks with the given activations.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that, for any fixed T>0 and p in (0,infty), solutions u_d of semilinear heat equations with Lipschitz nonlinearities on [0,T] x R^d can be approximated at time T in the L^p norm by DNNs using ReLU, leaky ReLU or softplus activations, with the number of parameters growing at most polynomially in d and 1/epsilon, provided the initial data functions admit such DNN approximations without the curse of dimensionality. This extends earlier results that were restricted to the L^2 norm and the ReLU activation.

Significance. If the proofs hold, the result supplies a clean technical extension of known approximation-theoretic guarantees for DNNs applied to high-dimensional Kolmogorov PDEs. Broadening the admissible activations and the range of p strengthens the theoretical case that DNNs can overcome the curse of dimensionality for this class of PDEs, conditional on the initial-data hypothesis that is already standard in the literature.

minor comments (2)
  1. The dependence of the constants on p and on the Lipschitz constant of the nonlinearity should be stated explicitly in the main theorem statement to make the polynomial-in-d claim fully transparent.
  2. Notation for the DNN parameter count N(d,epsilon) is used before it is formally defined; a forward reference or early definition would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive summary, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged · score 2.0 of 10

Conditional result on external initial-data assumption; minor self-citations not load-bearing

full rationale

The paper establishes a conditional statement that terminal-time L^p approximability by DNNs (ReLU/leaky ReLU/softplus) follows from the assumption that initial-value functions can be approximated without the COD. This generalizes prior L^2/ReLU results via technical extension of error-propagation arguments rather than any internal reduction of the target quantity to a fitted parameter or self-defined input. The weakest assumption is explicitly external and not derived within the paper. Self-citations to earlier works appear but are not load-bearing for the new generalization, satisfying the criteria for at most minor circularity.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The proof rests on standard properties of the listed activation functions and on the external assumption that initial data admit DNN approximations without COD; no new free parameters or invented entities are introduced.

assumptions (2)
  • domain assumption ReLU, leaky ReLU and softplus satisfy the approximation-theoretic properties used in the prior L^2 result
    Invoked to extend the earlier theorem to the new activations.
  • domain assumption Solutions of the semilinear heat PDE with Lipschitz nonlinearity admit the regularity needed for the approximation argument
    Standard background fact for Kolmogorov PDEs.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense." pith.science (2026). https://pith.science/paper/2309.13722

@misc{pith2026230913722,
  author       = {Pith},
  title        = {Pith review of: Deep neural networks with ReLU, leaky ReLU, and softplus activation provably overcome the curse of dimensionality for Kolmogorov partial differential equations with Lipschitz nonlinearities in the $L^p$-sense},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2309.13722}},
  note         = {Machine review of arXiv:2309.13722}
}
abstract

Recently, several deep learning (DL) methods for approximating high-dimensional partial differential equations (PDEs) have been proposed. The interest that these methods have generated in the literature is in large part due to simulations which appear to demonstrate that such DL methods have the capacity to overcome the curse of dimensionality (COD) for PDEs in the sense that the number of computational operations they require to achieve a certain approximation accuracy $\varepsilon\in(0,\infty)$ grows at most polynomially in the PDE dimension $d\in\mathbb N$ and the reciprocal of $\varepsilon$. While there is thus far no mathematical result that proves that one of such methods is indeed capable of overcoming the COD, there are now a number of rigorous results in the literature that show that deep neural networks (DNNs) have the expressive power to approximate PDE solutions without the COD in the sense that the number of parameters used to describe the approximating DNN grows at most polynomially in both the PDE dimension $d\in\mathbb N$ and the reciprocal of the approximation accuracy $\varepsilon>0$. Roughly speaking, in the literature it is has been proved for every $T>0$ that solutions $u_d\colon [0,T]\times\mathbb R^d\to \mathbb R$, $d\in\mathbb N$, of semilinear heat PDEs with Lipschitz continuous nonlinearities can be approximated by DNNs with ReLU activation at the terminal time in the $L^2$-sense without the COD provided that the initial value functions $\mathbb R^d\ni x\mapsto u_d(0,x)\in\mathbb R$, $d\in\mathbb N$, can be approximated by ReLU DNNs without the COD. It is the key contribution of this work to generalize this result by establishing this statement in the $L^p$-sense with $p\in(0,\infty)$ and by allowing the activation function to be more general covering the ReLU, the leaky ReLU, and the softplus activation functions as special cases.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionality

    math.OC 2025-06 accept novelty 6.0 of 10

    Q-functions of infinite-horizon discounted MDPs with finite action sets are approximable by leaky ReLU networks with polynomially growing parameter counts, provided rewards and transitions are themselves DNN-approximable.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages · cited by 1 Pith paper

  1. [1]

    Approximation properties of residual neural networks for Kolmogorov PDEs.Discrete Contin

    Baggenstos, J., and Salimov a, D. Approximation properties of residual neural networks for Kolmogorov PDEs.Discrete Contin. Dyn. Syst. Ser. B 28, 5 (2023), 3193– 3215

    work page 2023

  2. [2]

    Numerical solution of inverse problems by weak adversarial networks.Inverse Problems 36, 11 (2020), 115003, 31

    Bao, G., Ye, X., Zang, Y., and Zhou, H. Numerical solution of inverse problems by weak adversarial networks.Inverse Problems 36, 11 (2020), 115003, 31

    work page 2020

  3. [3]

    Deep splitting method for parabolic PDEs

    Beck, C., Becker, S., Cheridito, P., Jentzen, A., and Neufeld, A. Deep splitting method for parabolic PDEs. SIAM J. Sci. Comput. 43, 5 (2021), A3135– A3154. 47

    work page 2021

  4. [4]

    Solving the Kolmogorov PDE by means of deep learning.J

    Beck, C., Becker, S., Grohs, P., Jaaf ari, N., and Jentzen, A. Solving the Kolmogorov PDE by means of deep learning.J. Sci. Comput. 88, 3 (2021), Paper No. 73, 28

    work page 2021

  5. [5]

    Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order back- ward stochastic differential equations.J

    Beck, C., E, W., and Jentzen, A. Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order back- ward stochastic differential equations.J. Nonlinear Sci. 29, 4 (2019), 1563–1619

    work page 2019

  6. [6]

    On existence and uniqueness properties for solutions of stochastic fixed point equations.Discrete Contin

    Beck, C., Gonon, L., Hutzenthaler, M., and Jentzen, A. On existence and uniqueness properties for solutions of stochastic fixed point equations.Discrete Contin. Dyn. Syst. Ser. B 26, 9 (2021), 4927–4962

    work page 2021

  7. [7]

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

    Beck, C., Gonon, L., and Jentzen, A. Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations. Partial Differ. Equ. Appl. 5, 6 (2024), Paper No. 31, 47

    work page 2024

  8. [8]

    Beck, C., Hornung, F., Hutzenthaler, M., Jentzen, A., and Kruse, T. Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard ap- proximations. J. Numer. Math. 28, 4 (2020), 197–222

    work page 2020

Show all 65 references
  1. [9]

    On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations

    Beck, C., Hutzenthaler, M., and Jentzen, A. On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations. Stochastics and Dynamics 21, 08 (2021), 2150048

    2021

  2. [10]

    An overview on deep learning-based approximation methods for partial differential equations.Discrete Contin

    Beck, C., Hutzenthaler, M., Jentzen, A., and Kuckuck, B. An overview on deep learning-based approximation methods for partial differential equations.Discrete Contin. Dyn. Syst. Ser. B 28, 6 (2023), 3697–3746

    2023

  3. [11]

    NumericalsimulationsforfullhistoryrecursivemultilevelPicard approximations for systems of high-dimensional partial differential equations.Commun

    Becker, S., Braunw arth, R., Hutzenthaler, M., Jentzen, A., and von Wurstemberger, P. NumericalsimulationsforfullhistoryrecursivemultilevelPicard approximations for systems of high-dimensional partial differential equations.Commun. Comput. Phys. 28, 5 (2020), 2109–2138

    2020

  4. [12]

    Dynamic programming

    Bellman, R. Dynamic programming. Princeton Landmarks in Mathematics. Prince- ton University Press, Princeton, NJ, 2010. Reprint of the 1957 edition, With a new introduction by Stuart Dreyfus

    2010

  5. [13]

    A unified deep artificial neural network approach to partial differential equations in complex geometries.Neurocomputing 317 (2018), 28– 41

    Berg, J., and Nyström, K. A unified deep artificial neural network approach to partial differential equations in complex geometries.Neurocomputing 317 (2018), 28– 41

    2018

  6. [14]

    Berner, J., Grohs, P., and Jentzen, A. 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. SIAM J. Math. Dat...

    2020

  7. [15]

    Blechschmidt, J., and Ernst, O. G. Three ways to solve partial differential equa- tionswithneuralnetworks—areview. GAMM-Mitt. 44, 2(2021), PaperNo.e202100006, 29

    2021

  8. [16]

    Machine learning for semi linear PDEs

    Chan-W ai-Nam, Q., Mikael, J., and W arin, X. Machine learning for semi linear PDEs. J. Sci. Comput. 79, 3 (2019), 1667–1712

    2019

  9. [17]

    Deep Runge–Kutta schemes for BSDEs

    Chassagneux, J.-F., Chen, J., and Frikha, N. Deep Runge–Kutta schemes for BSDEs. arXiv:2212.14372 (2022), 33 pages

    2022

  10. [18]

    A., Hutzenthaler, M., and Werner, P

    Cioica-Licht, P. A., Hutzenthaler, M., and Werner, P. T. Deep neural net- works overcome the curse of dimensionality in the numerical approximation of semilinear partial differential equations.arXiv:2205.14398 (2022), 34 pages

    2022

  11. [19]

    Deep learning-based numerical methods for high- dimensional parabolic partial differential equations and backward stochastic differential equations

    E, W., Han, J., and Jentzen, A. Deep learning-based numerical methods for high- dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat. 5, 4 (2017), 349–380

    2017

  12. [20]

    Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning.Nonlinearity 35, 1 (2022), 278–310

    E, W., Han, J., and Jentzen, A. Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning.Nonlinearity 35, 1 (2022), 278–310

    2022

  13. [21]

    E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T. On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations.J. Sci. Comput. 79, 3 (2019), 1534–1571

    2019

  14. [22]

    Multilevel Picard itera- tions for solving smooth semilinear parabolic heat equations.Partial Differ

    E, W., Hutzenthaler, M., Jentzen, A., and Kruse, T. Multilevel Picard itera- tions for solving smooth semilinear parabolic heat equations.Partial Differ. Equ. Appl. 2, 6 (2021), Paper No. 80, 31

    2021

  15. [23]

    The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems.Commun

    E, W., and Yu, B. The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems.Commun. Math. Stat. 6, 1 (2018), 1–12

    2018

  16. [24]

    DNN expression rate analysis of high-dimensional PDEs: Application to option pricing.Constr

    Elbrächter, D., Grohs, P., Jentzen, A., and Schw ab, C. DNN expression rate analysis of high-dimensional PDEs: Application to option pricing.Constr. Approx. (2021), 1–69

    2021

  17. [25]

    Fractional weak adversarial networks for the sta- tionary fractional advection dispersion equations.Z

    Feng, D., Yang, Z., and Zou, S. Fractional weak adversarial networks for the sta- tionary fractional advection dispersion equations.Z. Angew. Math. Phys. 75, 5 (2024), Paper No. 168, 20

    2024

  18. [26]

    Asymptotic expansion as prior knowledge in deep learning method for high dimensional BSDEs.Asia-Pacific Financial Markets (Mar 2019)

    Fujii, M., Takahashi, A., and Takahashi, M. Asymptotic expansion as prior knowledge in deep learning method for high dimensional BSDEs.Asia-Pacific Financial Markets (Mar 2019)

    2019

  19. [27]

    Approximation error analysis of some deep backward schemes for nonlinear PDEs.SIAM J

    Germain, M., Pham, H., and W arin, X. Approximation error analysis of some deep backward schemes for nonlinear PDEs.SIAM J. Sci. Comput. 44, 1 (2022), A28–A56. 49

    2022

  20. [28]

    Neural Networks–Based Algorithms for Stochastic Control and PDEs in Finance

    Germain, M., Pham, H., and W arin, X. Neural Networks–Based Algorithms for Stochastic Control and PDEs in Finance. Cambridge University Press, 2023, pp. 426– –452

    2023

  21. [29]

    B., Jentzen, A., and Welti, T

    Giles, M. B., Jentzen, A., and Welti, T. Generalised multilevel Picard approxi- mations. arXiv:1911.03188 (2019), 61 pages

    1911

  22. [30]

    Uniform er- ror estimates for artificial neural network approximations for heat equations.IMA J

    Gonon, L., Grohs, P., Jentzen, A., Kofler, D., and Šiška, D. Uniform er- ror estimates for artificial neural network approximations for heat equations.IMA J. Numer. Anal. 42, 3 (2022), 1991–2054

    2022

  23. [31]

    Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models.Finance Stoch

    Gonon, L., and Schw ab, C. Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models.Finance Stoch. 25, 4 (2021), 615–657

    2021

  24. [32]

    Deep neural network approximation for high- dimensional parabolic Hamilton–Jacobi–Bellman equations.arXiv:2103.05744 (2021), 23 pages

    Grohs, P., and Herrmann, L. Deep neural network approximation for high- dimensional parabolic Hamilton–Jacobi–Bellman equations.arXiv:2103.05744 (2021), 23 pages

    2021

  25. [33]

    Deep neural network approximation for high- dimensional elliptic PDEs with boundary conditions.IMA J

    Grohs, P., and Herrmann, L. Deep neural network approximation for high- dimensional elliptic PDEs with boundary conditions.IMA J. Numer. Anal. 42, 3 (2022), 2055–2082

    2022

  26. [34]

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

    Grohs, P., Hornung, F., Jentzen, A., and von Wurstemberger, P. A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations.Mem. Amer. Math. Soc. 284, 1410 (2023), v+93

    2023

  27. [35]

    Space-time er- ror estimates for deep neural network approximations for differential equations.Adv

    Grohs, P., Hornung, F., Jentzen, A., and Zimmermann, P. Space-time er- ror estimates for deep neural network approximations for differential equations.Adv. Comput. Math. 49, 1 (2023), Paper No. 4, 78

    2023

  28. [36]

    Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms.Partial Differ

    Grohs, P., Jentzen, A., and Salimov a, D. Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms.Partial Differ. Equ. Appl. 3, 4 (2022), Paper No. 45, 41

    2022

  29. [37]

    Solving high-dimensional partial differential equa- tions using deep learning.Proc

    Han, J., Jentzen, A., and E, W. Solving high-dimensional partial differential equa- tions using deep learning.Proc. Natl. Acad. Sci. USA 115, 34 (2018), 8505–8510

    2018

  30. [38]

    Convergence of the deep BSDE method for coupled FBSDEs

    Han, J., and Long, J. Convergence of the deep BSDE method for coupled FBSDEs. Probab. Uncertain. Quant. Risk 5(2020), Paper No. 5, 33

    2020

  31. [39]

    Deep Primal-Dual Algorithm for BSDEs: Applications of Machine Learning to CVA and IM

    Henry-Labordère, P. Deep Primal-Dual Algorithm for BSDEs: Applications of Machine Learning to CVA and IM. (November 15, 2017), 16 pages. Available at SSRN: https://ssrn.com/abstract=3071506

    2017

  32. [40]

    A., and Johnson, C

    Horn, R. A., and Johnson, C. R. Matrix analysis. Cambridge University Press, Cambridge, 1985. 50

    1985

  33. [41]

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

    Hornung, F., Jentzen, A., and Salimov a, D. Space-time deep neural network approximations for high-dimensional partial differential equations. arXiv:2006.02199 (2020), 52 pages. Accepted in J. Comput. Math

    2006

  34. [42]

    Hu, Z., Ka w aguchi, K., Zhang, Z., and Karniadakis, G. E. Tackling the curse of dimensionality in fractional and tempered fractional PDEs with physics-informed neural networks. Comput. Methods Appl. Mech. Engrg. 432(2024), Paper No. 117448, 13

    2024

  35. [43]

    Deep backward schemes for high-dimensional nonlinear PDEs

    Huré, C., Pham, H., and W arin, X. Deep backward schemes for high-dimensional nonlinear PDEs. Math. Comp. 89, 324 (2020), 1547–1579

    2020

  36. [44]

    Overcoming the curse of dimen- sionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities.Found

    Hutzenthaler, M., Jentzen, A., and Kruse, T. Overcoming the curse of dimen- sionality in the numerical approximation of parabolic partial differential equations with gradient-dependent nonlinearities.Found. Comput. Math. 22, 4 (2022), 905–966

    2022

  37. [45]

    Hutzenthaler, M., Jentzen, A., Kruse, T., and Nguyen, T. A. Multilevel Pi- card approximations for high-dimensional semilinear second-order PDEs with Lipschitz nonlinearities. arXiv:2009.02484 (2020), 37 pages

    2009

  38. [46]

    Hutzenthaler, M., Jentzen, A., Kruse, T., and Nguyen, T. A. A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations. Partial Differ. Equ. Appl. 1, 2 (2020), Paper No. 10, 34

    2020

  39. [47]

    A., and von Wurstemberger, P

    Hutzenthaler, M., Jentzen, A., Kruse, T., Nguyen, T. A., and von Wurstemberger, P. Overcoming the curse of dimensionality in the numerical ap- proximation of semilinear parabolic partial differential equations.Proc. A. 476, 2244 (2020), 20190630, 25

    2020

  40. [48]

    Hutzenthaler, M., Jentzen, A., Kuckuck, B., and Padgett, J. L. Strong Lp- error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations. arXiv:2110.08297 (2021), 42 pages. Revision requested from Numerical Algorithms

    2021

  41. [49]

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

    Hutzenthaler, M., Jentzen, A., and von Wurstemberger, P. Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks. Electron. J. Probab. 25(2020), Paper No. 101, 73

    2020

  42. [50]

    Multilevel Picard approximations of high- dimensional semilinear parabolic differential equations with gradient-dependent nonlin- earities

    Hutzenthaler, M., and Kruse, T. Multilevel Picard approximations of high- dimensional semilinear parabolic differential equations with gradient-dependent nonlin- earities. SIAM J. Numer. Anal. 58, 2 (2020), 929–961

    2020

  43. [51]

    Hutzenthaler, M., Kruse, T., and Nguyen, T. A. Multilevel Picard approxima- tions for McKean-Vlasov stochastic differential equations.J. Math. Anal. Appl. 507, 1 (2022), Paper No. 125761, 14

    2022

  44. [52]

    Hutzenthaler, M., and Nguyen, T. A. Multilevel Picard approximations of high- dimensional semilinear partial differential equations with locally monotone coefficient functions. Appl. Numer. Math. 181(2022), 151–175. 51

    2022

  45. [53]

    Deep curve-dependent PDEs for affine rough volatility

    Jacquier, A., and Oumgari, M. Deep curve-dependent PDEs for affine rough volatility. SIAM J. Financial Math. 14, 2 (2023), 353–382

    2023

  46. [54]

    Jentzen, A., Salimov a, D., and Welti, T. A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kol- mogorov partial differential equations with constant diffusion and nonlinear drift coef- ficients. Commun. Math. S...

    2021

  47. [55]

    A theoretical analysis of deep neural networks and parametric PDEs.Constr

    Kutyniok, G., Petersen, P., Raslan, M., and Schneider, R. A theoretical analysis of deep neural networks and parametric PDEs.Constr. Approx. 55, 1 (2022), 73–125

    2022

  48. [56]

    Multilevel Picard approximation algorithm for semilin- ear partial integro-differential equations and its complexity analysis.arXiv:2205.09639 (2022), 54 pages

    Neufeld, A., and Wu, S. Multilevel Picard approximation algorithm for semilin- ear partial integro-differential equations and its complexity analysis.arXiv:2205.09639 (2022), 54 pages

    2022

  49. [57]

    Tractability of multivariate problems

    Nov ak, E., and Woźniakowski, H. Tractability of multivariate problems. Vol. 1: Linear information, vol. 6 of EMS Tracts in Mathematics. European Mathematical Society (EMS), Zürich, 2008

    2008

  50. [58]

    Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space.Partial Differ

    Nüsken, N., and Richter, L. Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space.Partial Differ. Equ. Appl. 2, 4 (2021), Paper No. 48, 48

    2021

  51. [59]

    Neural networks-based backward scheme for fully nonlinear PDEs.Partial Differ

    Pham, H., W arin, X., and Germain, M. Neural networks-based backward scheme for fully nonlinear PDEs.Partial Differ. Equ. Appl. 2, 1 (2021), Paper No. 16, 24

    2021

  52. [60]

    Raissi, M., Perdikaris, P., and Karniadakis, G. E. Physics–informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.J. Comput. Phys. 378(2019), 686–707

    2019

  53. [61]

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

    Reisinger, C., and Zhang, Y. Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems. Anal. Appl. (Singap.) 18, 6 (2020), 951–999

    2020

  54. [62]

    Simon, M. K. Probability distributions involving Gaussian random variables: A hand- book for engineers and scientists. Springer Science & Business Media, 2007

    2007

  55. [63]

    DGM: A deep learning algorithm for solving partial differential equations.J

    Sirignano, J., and Spiliopoulos, K. DGM: A deep learning algorithm for solving partial differential equations.J. Comput. Phys. 375(2018), 1339–1364

    2018

  56. [64]

    Towards fast weak adversarial trainingtosolvehighdimensionalparabolicpartialdifferentialequationsusingXNODE- WAN

    V alsecchi Oliv a, P., Wu, Y., He, C., and Ni, H. Towards fast weak adversarial trainingtosolvehighdimensionalparabolicpartialdifferentialequationsusingXNODE- WAN. J. Comput. Phys. 463(2022), Paper No. 111233, 17

    2022

  57. [65]

    Weak adversarial networks for high- dimensional partial differential equations.J

    Zang, Y., Bao, G., Ye, X., and Zhou, H. Weak adversarial networks for high- dimensional partial differential equations.J. Comput. Phys. 411(2020), 109409, 14. 52

    2020

Pith tools

Reviewed May 24, 2026 · model on record in the stance chip above.