Deep Truncated FBSDE Method for High-Dimensional Nonlinear PDEs and Fully Coupled FBSDEs

Xinyu Cheng; Yunzhang Li; Weiran Xiong

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Deep Truncated FBSDE Method for High-Dimensional Nonlinear PDEs and Fully Coupled FBSDEs

Xinyu Cheng, Yunzhang Li, Weiran Xiong

arXiv:2609.39616v2Sep 30, 2026math.NAcs.NAmath.PR

TL;DR

Fully coupled FBSDE solvers can become unstable because parameter-dependent forward paths feed errors back into optimization. The paper proposes a deep truncated FBSDE method that combines iterative decoupling with gradient truncation, and reports improved accuracy and stability across low- and high-dimensional experiments.

Problem
In fully coupled FBSDEs, dependence of forward coefficients on backward variables makes simulated paths parameter-dependent, potentially destabilizing training, especially under strong coupling or long horizons.
Source: §1 · p. 1
Method
The method combines gradient-truncated iterative decoupling and fictitious-play averaging to retain coupled simulated dynamics while reducing gradient feedback through parameter-dependent forward paths.
Source: §1.1 · p. 2
Results
Experiments report improved accuracy and stability in low- and high-dimensional comparisons, including for degenerate convection-dominated equations.
Source: §1.1 · p. 2
Takeaways & Limitations
The coupled formulation provides a stable representation of nonlinear transport for degenerate convection-dominated equations without singular generator terms.
Source: §1.1 · p. 2
The theoretical analysis is conditional on its assumptions, including a unique classical PDE solution and a unique adapted FBSDE solution.
Source: §1 · p. 2, §5.1 · p. 11

Abstract

from arXiv · show

In this paper, we introduce a deep truncated forward-backward stochastic differential equation (FBSDE) method for high-dimensional partial differential equations (PDEs) and fully coupled FBSDEs. The proposed method combines gradient-truncated iterative decoupling with fictitious-play averaging to separate the forward and backward processes in a coupled framework. This preserves the coupled dynamics while reducing the unstable feedback induced by parameter-dependent forward paths during optimization. Furthermore, we incorporate a pathwise consistency term to create explicit local gradient shortcuts, thereby providing a structural mechanism that may mitigate gradient vanishing. We also derive a residual-based error estimate and establish conditional convergence of the fully discrete numerical approximations under suitable conditions, in which the pathwise consistency loss is not required. Our approach is particularly effective for convection-dominated equations, where the coupled formulation provides a stable representation of nonlinear transport without introducing singular terms into the BSDE. Numerical experiments demonstrate improved accuracy and stability in both low- and high-dimensional problems and robust performance for strongly coupled problems.

1 Introduction

The introduction frames the challenge of unstable training for fully coupled FBSDEs and presents a deep truncated method for high-dimensional quasilinear PDEs. It summarizes the method’s theoretical guarantees, numerical performance, related work, and paper organization.

1.1 Main results and contributions

The deep truncated FBSDE method targets high-dimensional quasilinear PDEs and corresponding fully coupled FBSDEs, using gradient-truncated iterative decoupling and fictitious-play averaging.It retains coupling in simulated dynamics while reducing gradient feedback through parameter-dependent forward trajectories; pathwise consistency adds intermediate-time constraints.Source: §1 · p. 1, §1.1 · p. 2
The paper establishes a residual-based error estimate and conditional convergence under suitable assumptions, neither of which requires the pathwise consistency term.The analysis also identifies local gradient contributions from pathwise consistency that may mitigate gradient vanishing over long horizons.Source: §1.1 · p. 2
Experiments in low and high dimensions report improved accuracy and stability, including for degenerate convection-dominated equations where the coupled formulation avoids singular generator terms.The paper also reports competitive computational cost and says ablations support the stabilizing role of gradient truncation.Source: §1.1 · p. 2

1.2 Related works

Prior deep FBSDE approaches differ in how they handle forward–backward coupling during numerical simulation and optimization, a distinction central to this method’s design.The introduction notes that some related schemes are designed primarily for decoupled FBSDEs or impose restrictions on coefficient dependence.Source: §1.2 · pp. 2–3

1.3 Outline of the paper

The paper proceeds from existing methods and motivation to the proposed method, gradient analysis, error and convergence results, numerical experiments, and conclusions.The method appears in Section 3, theory in Sections 4–5, and experiments in Section 6.Source: §1.3 · p. 3

2 Existing methods and motivation

This section reviews deep FBSDE methods and motivates a strategy for managing coupling during optimization. It contrasts parameter-independent decoupled paths with fully coupled, parameter-dependent paths, then introduces iterative decoupling and gradient truncation as responses to their feedback instability.

2.1 Notation

The setup uses a filtered probability space with Brownian motion and defines norms, time partitions, and discrete approximations for the stochastic processes.The paper denotes discrete approximations of X and Y at time levels and of Z up to the preceding time level.Source: §2.1 · p. 3

2.2 Deep FBSDE methods

In decoupled FBSDEs, the deep BSDE method uses a parameter-independent forward trajectory, so parameter updates act through the backward recursion.For fully coupled systems, forward paths depend on approximations of Y and Z, and parameter updates change the trajectories used to evaluate terminal loss.Source: §2.2 · pp. 3–4

2.3 Motivation: breaking the vicious cycle

Parameter-dependent paths can form a feedback loop: inaccurate backward approximations distort forward trajectories, while optimization along those paths may further amplify errors.The paper says this instability can be particularly pronounced under strong coupling or long time horizons.Source: §2.3 · p. 4
Iterative decoupling evaluates the BSDE along a fixed reference path updated by fictitious-play averaging, while gradient truncation blocks gradient propagation through the forward path.The backward variables still numerically influence forward dynamics, preserving coupling in simulation while reducing parameter-dependent feedback during optimization.Source: §2.3 · p. 4

3 Deep truncated FBSDE method

The method iteratively decouples the forward and backward processes while retaining their coupled influence in simulated dynamics. Its numerical scheme combines reference-path averaging, Feynman–Kac reconstruction, and pathwise consistency optimization.

3.1 Gradient-truncated decoupling with fictitious-play averaging

Gradient-truncated decoupling evaluates the backward equation along a frozen reference path, uses its solution to generate the current forward trajectory, and updates the reference by running averages.The average is chosen for compatibility with convergence analysis and may stabilize the reference path relative to a Picard update using only the latest iterate.Source: §3.1 · p. 5

3.2 Feynman–Kac reconstruction along the reference path

Automatic differentiation reconstructs Z from the neural representation of Y along the reference path, avoiding a separate neural parametrization for Z.This retains a derivative-based relation between the Y and Z approximations along the trajectory.Source: §3.2 · p. 5

3.3 Pathwise consistent optimization over long horizons

The pathwise consistency loss penalizes backward-process discrepancies over the horizon, while the terminal target remains evaluated at the current forward state.Its intermediate-time local gradient contributions bypass the full terminal-loss propagation chain and may mitigate gradient vanishing over long horizons.Source: §3.3 · p. 5

3.4 Numerical scheme

The discrete scheme applies stop-gradient to backward variables in the forward dynamics, preserving their numerical values while blocking parameter gradients through simulated forward trajectories.The forward equation uses Euler–Maruyama, and the backward equation is evaluated recursively forward in time.Source: §3.4 · p. 6
The same Brownian batch is reused across inner decoupling steps, and pathwise deviations in both forward and backward trajectories inform the stopping criterion.The number of decoupling steps trades additional refinement against additional forward–backward updates; for decoupled FBSDEs, the method sets P = 1 and omits truncation and averaging.Source: §3.4 · pp. 7–8

4 Gradient propagation and gradient shortcuts

The analysis decomposes gradients into local pathwise-consistency contributions and terms propagated through later time levels. Under stated regularity assumptions, terminal-loss gradients can attenuate over long horizons, while the local contributions bypass the full propagation chain.

4.1 Assumptions and main theorem

Under coefficient regularity and neural-network differentiability assumptions, Theorem 4.1 decomposes the total-loss gradient into local pathwise-consistency contributions and a propagated remainder.The local terms do not involve the full propagation chain; the remainder collects later-time and terminal-loss contributions.Source: §4.1 · p. 8

4.2 Gradient propagation and attenuation

For early-time parameters, terminal-loss gradients propagate through a product of matrices whose norm is exponentially bounded in the effective horizon under the propagation-damping assumption.Consequently, terminal-loss gradients may become weak over long horizons.Source: §4.2 · p. 9

4.3 Local gradient shortcuts from pathwise consistency

The pathwise consistency loss adds local gradient contributions at neighboring time levels that do not contain the full temporal propagation chain.These contributions complement the terminal-loss gradient.Source: §4.3 · p. 10

5 Error estimates and convergence

This section provides residual-based error bounds and conditional convergence results for the FBSDE scheme, showing contraction of the exact decoupling iteration and convergence of fully discrete approximations under the stated assumptions; the analysis excludes Monte Carlo sampling error and stochastic-optimization convergence.

5 Error estimates and convergence

The analysis uses exact expectations and only the terminal loss, excluding Monte Carlo sampling error and stochastic-optimization convergence; strongly coupled problems are studied numerically elsewhere.The convergence theorem itself requires weak coupling.Source: §5 · p. 10

5.1 Notations, assumptions, and main results

Theorem 5.1 provides a residual-based error estimate, using terminal loss, terminal reference mismatch, and reference-path discrepancy.The analysis compares the discrete process with the original coupled FBSDE solution; the residual quantities distinguish terminal mismatch from discrepancy between discrete and continuous reference paths.Source: §5.1 · pp. 11–12
Theorem 5.2 establishes conditional convergence of the fully discrete scheme under uniform-moment, weak-coupling, and residual-consistency assumptions.Weak coupling requires KF KB < 1; the residual-consistency assumption concerns the terminal loss and terminal reference mismatch as iteration count grows and the time mesh is refined.Source: §5.1 · p. 12

5.2 Proof of the error estimate

The proof develops mesh-independent moment bounds, continuous-time extension estimates, stability estimates, and a recursion for reference-path discrepancy.These intermediate estimates support the residual-based error analysis for the auxiliary decoupling systems.Source: §5.2 · pp. 13–16
The proof establishes the baseline error estimate (5.9) and then derives a refined estimate using ζ∗ = √(1 + ζ) − 1.The baseline estimate follows by taking η = 1; the refined estimate uses η = ζ∗ and Lemma 5.5.Source: §5.2 · p. 20

5.3 Proof of the convergence

Under Assumptions 5.1–5.4, the refined error estimates hold uniformly over decoupling steps p ≥ 1 and partitions π.For every ζ > 0, the estimates use constants independent of p and π, and the theorem’s constants Λp can be chosen uniformly bounded.Source: §5.3 · p. 20
The exact decoupling iteration converges because q = √(KBKF) < 1, so its decoupling error tends to zero as p → ∞.The lemma assumes Assumptions 5.1, 5.2, and 5.5.Source: §5.3 · p. 21
Under the theorem’s stated assumptions, the fully discrete convergence proof shows the relevant error terms vanish as the iteration count grows and the discretization is refined.The proof uses the contraction condition qζ < 1 and concludes that the three terms in the final bound vanish.Source: §5.3 · p. 22

6 Numerical experiments

Experiments evaluate the method on Burgers, Allen–Cahn, and gene-flow problems, from two to 100 dimensions and across shock regimes and longer horizons. The reported results show strong accuracy and stability across these settings, with case-specific comparisons below.

6.1 Example 1: scalar Burgers equation

At T = 1.5 and ε = 10−8, both L2 and L∞ errors are of order 10−6, showing stable approximation in the nearly inviscid post-shock regime.The coupled formulation avoids the singular factor ε−1/2 in the decoupled generator.Source: §6.1 · pp. 23–24, Table 1 · p. 24

6.2 Example 2: 100-dimensional spatially heterogeneous Burgers-type equation

The proposed method achieves a mean relative error close to 1% on the 100-dimensional Burgers-type problem, while competing methods are unreliable or unstable despite heavier DBDP training.Figure 5 tracks training relative error and numerical-solution convergence across ten independent runs.Source: §6.2 · p. 26, Figure 5 · p. 26

6.3 Example 3: two-dimensional Allen–Cahn equation

For both ring and seven-disk initial conditions, the method captures Allen–Cahn interfaces and achieves the smallest reported relative errors, while several alternatives are numerically unstable.All methods use the decoupled formulation in this experiment.Source: §6.3 · pp. 26–27, Table 3 · p. 27

6.4 Example 4: 100-dimensional Allen–Cahn equations √

4.95 × 10−4 mean relative error on the double-well test; the method has the lowest mean relative error in both tests and training times comparable to the fastest methods.For the logarithmic-potential test, its relative error reaches 3.24 × 10−3.Source: Table 4 · p. 29, §6.4 · p. 29

6.5 Example 5: gene flow model

At T = 3, 5, and 7, the method stays close to reference solutions, captures release-region expansion and merging, and has the smallest relative L2 and L∞ errors in every tested case.The competing coupled deep BSDE and Ji et al. methods deteriorate as T increases.Source: Table 6 · p. 30, §6.5 · pp. 30–31, Figure 8 · p. 31

7 Conclusions and future work

The deep truncated FBSDE method combines gradient-truncated iterative decoupling, fictitious-play averaging, and pathwise consistency optimization for high-dimensional nonlinear PDEs and fully coupled FBSDEs. The paper reports residual-based error estimates, conditional convergence under suitable assumptions, and improved numerical accuracy and stability, while identifying extensions for future work.

7 Conclusions and future work

The deep truncated FBSDE method combines gradient-truncated iterative decoupling, fictitious-play averaging, and pathwise consistency optimization to improve training stability.The framework targets high-dimensional nonlinear PDEs and fully coupled FBSDEs.Source: §7 · p. 31
The paper establishes residual-based error estimates and conditional convergence under suitable assumptions, and reports improved accuracy and stability in experiments including strongly coupled and degenerate convection-dominated problems.Source: §7 · p. 31
Future work will extend convergence analysis to stronger coupling regimes, quantify neural approximation, sampling, and optimization errors, and explore broader classes of nonlinear PDEs and stochastic systems.Source: §7 · p. 31