mlsanityData to 7 Oct 2026

Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs

arXiv 2405.05192 ↗First posted 8 May 2024math.NA cs.LG cs.NAcited by 12

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Abstract (arXiv)

In this paper, we present a randomized extension of the deep splitting algorithm introduced in [Beck, Becker, Cheridito, Jentzen, and Neufeld (2021)] using random neural networks suitable to approximately solve both high-dimensional nonlinear parabolic PDEs and PIDEs with jumps having (possibly) infinite activity. We provide a full error analysis of our so-called random deep splitting method. In particular, we prove that our random deep splitting method converges to the (unique viscosity) solution of the nonlinear PDE or PIDE under consideration. Moreover, we empirically analyze our random deep splitting method by considering several numerical examples including both nonlinear PDEs and nonlinear PIDEs relevant in the context of pricing of financial derivatives under default risk. In particular, we empirically demonstrate in all examples that our random deep splitting method can approximately solve nonlinear PDEs and PIDEs in 10'000 dimensions within seconds.

Key facts. Cited by 12 papers in the map (data through 7 Oct 2026). Builds on 37 of its 117 references that are in the map. First posted on arXiv 8 May 2024.

Builds on

37 of 117 references.

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25 more
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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 3; 2025: 3; 2026: 6.

12 papers.

  1. 2024Solving stochastic partial differential equations using neural networks in the Wiener chaos expansioncited by 6
  2. 2025Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programscited by 5
  3. 2024Multilevel 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-sensecited by 3
  4. 2024RandNet-Parareal: a time-parallel PDE solver using Random Neural Networkscited by 3
  5. 2025Error analysis for the deep Kolmogorov methodcited by 0
  6. 2025Robust Exploratory Stopping under Ambiguity in Reinforcement Learningcited by 0
  7. 2026INEUS: Iterative Neural Solver for High-Dimensional PIDEscited by 0
  8. 2026Unbiased and Second-Order-Free Training for High-Dimensional PDEscited by 0
  9. 2026Random Neural Network Expressivity for Non-Linear Partial Differential Equationscited by 0
  10. 2026Neural operators approximate strongly continuous convex monotone semigroupscited by 0
  11. 2026Spectral Convergence of Random Feature Method in Multiple Dimensionscited by 0
  12. 2026Residual neural networks overcome the curse of dimensionality for semilinear heat equationscited by 0