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A deep backward regression-based scheme for high-dimensional nonlinear partial differential equations

Qiang Han, Shaolin Ji, Yunzhang Li

Abstract

A deep backward regression-based (DBR) scheme for solving high-dimensional nonlinear parabolic partial differential equations is proposed. Building upon the established DBDP method (Huré et al., 2020), our algorithm introduces a reformulation of the local loss functions that are sequentially optimized via backward induction at each time step. The core of this approach involves reformulating simulated backward stochastic difference equations into their conditional expectation representations, thereby recasting a projection-based stochastic optimization problem as a deterministic function-approximation task. By explicitly incorporating conditional expectations, the DBR scheme facilitates a denoising mechanism prior to loss evaluation. This architecture substantially mitigates numerical variance, resulting in enhanced training stability and superior generalization performance. Numerical results demonstrate that the DBR scheme consistently outperforms the DBDP1 method, maintaining accuracy up to d=200 for bounded solutions (see Table 1). Notably, for complex unbounded PDEs where the DBDP1 method fails beyond d=10, the DBR scheme remains robust up to $d=20$ with relative errors under 9.7% (see Table 6}). Theoretically, we derive rigorous upper error bounds and establish half-order convergence for the proposed scheme. Extensions to variational inequalities are also provided.

A deep backward regression-based scheme for high-dimensional nonlinear partial differential equations

Abstract

A deep backward regression-based (DBR) scheme for solving high-dimensional nonlinear parabolic partial differential equations is proposed. Building upon the established DBDP method (Huré et al., 2020), our algorithm introduces a reformulation of the local loss functions that are sequentially optimized via backward induction at each time step. The core of this approach involves reformulating simulated backward stochastic difference equations into their conditional expectation representations, thereby recasting a projection-based stochastic optimization problem as a deterministic function-approximation task. By explicitly incorporating conditional expectations, the DBR scheme facilitates a denoising mechanism prior to loss evaluation. This architecture substantially mitigates numerical variance, resulting in enhanced training stability and superior generalization performance. Numerical results demonstrate that the DBR scheme consistently outperforms the DBDP1 method, maintaining accuracy up to d=200 for bounded solutions (see Table 1). Notably, for complex unbounded PDEs where the DBDP1 method fails beyond d=10, the DBR scheme remains robust up to with relative errors under 9.7% (see Table 6}). Theoretically, we derive rigorous upper error bounds and establish half-order convergence for the proposed scheme. Extensions to variational inequalities are also provided.
Paper Structure (13 sections, 4 theorems, 98 equations, 2 figures, 6 tables, 1 algorithm)

This paper contains 13 sections, 4 theorems, 98 equations, 2 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Deep neural networks as function approximators
  4. Probabilistic representation for the solutions of PDEs
  5. Deep backward regression-based method
  6. Deep learning method to approximate the conditional expectation
  7. Error analysis
  8. Convergence analysis
  9. Extension to variational inequalities
  10. RDBR Scheme
  11. Convergence analysis
  12. Numerical experiments
  13. Conclusion

Key Result

Theorem 3.1

Suppose the assumptions (i)-(iv) hold. Let $(Y_{t_i},Z_{t_i})$ and $(\mathcal{Y}_{i},\mathcal{Z}_{i})$ be solutions of the FBSDEs (dFBSDE) and solutions of the DBR method (NS-DMC) respectively. Then, we have, for small enough $h$ where $C$ represents a positive generic constant which is independent of $\pi$ and may change from line to line.

Figures (2)

  • Figure 1: Estimated solution $\mathfrak{u}(t,x)$ obtained by DBR versus exact solution $u(t,x)$ for Example 1 with $d=1$.
  • Figure 2: Estimated solution $\overline{\mathfrak{u}}(t, x)$ obtained by DBR versus exact solution $u(t,x)$ for Example 2 with $d=1$.

Theorems & Definitions (4)

  • Theorem 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 4.3