Paper: arXiv 2212.14372

Authors: Jean-François Chassagneux, Junchao Chen, Noufel Frikha

Abstract

We propose a new probabilistic scheme which combines deep learning techniques with high order schemes for backward stochastic differential equations belonging to the class of Runge-Kutta methods to solve high-dimensional semi-linear parabolic partial differential equations. Our approach notably extends the one introduced in [Hure Pham Warin 2020] for the implicit Euler scheme to schemes which are more efficient in terms of discrete-time error. We establish some convergence results for our implemented schemes under classical regularity assumptions. We also illustrate the efficiency of our method for different schemes of order one, two and three. Our numerical results indicate that the Crank-Nicolson schemes is a good compromise in terms of precision, computational cost and numerical implementation.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework extending high-order Runge-Kutta schemes to deep neural network solvers for high-dimensional BSDEs and parabolic PDEs. It features strong theoretical convergence proofs alongside solid numerical validation for order 1, 2, and 3 schemes, though code availability is not explicitly specified.

Research Flowchart

  flowchart TD
    A["Research Goal: Solve High-Dim Parabolic PDEs via BSDEs"] --> B["Inputs: Terminal Conditions & High-Dim BSDE Formulations"]
    B --> C["Methodology: Deep Runge-Kutta Schemes"]
    C --> D["Computational Process: Train Neural Networks for Discretized RK Stages"]
    D --> E["Evaluation: Compare Order 1, 2, and 3 Schemes"]
    E --> F["Key Outcome: Convergence Proven & Crank-Nicolson Optimal for Cost/Precision"]