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Theoretical explanation of deep BSDE failure modes and deep multi-FBSDE success

Establish a theoretical characterization of the classes of coupled forward–backward stochastic differential equations for which the standard deep BSDE method fails and explain the mechanisms of this failure, and, in parallel, determine why the proposed deep multi-FBSDE method succeeds on these problems.

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Background

The authors present extensive numerical evidence that the deep multi-FBSDE method converges in scenarios where the standard deep BSDE method struggles or fails, including challenging coupled FBSDEs. However, they note that the reasons for these contrasting behaviors are not theoretically understood.

They call for a theoretical analysis that identifies problem classes where deep BSDE fails, explains why, and clarifies why deep multi-FBSDE works in such cases.

References

Although the method seems to work very well on challenging FBSDE, we still do not understand the reason for this and why the deep BSDE method fails. There is thus a need to both understand theoretically for what problems the deep BSDE does not work and why, and why the deep multi-FBSDE method does work for these problems.

— The deep multi-FBSDE method: a robust deep learning method for coupled FBSDEs (2503.13193 - Andersson et al., 17 Mar 2025) in Section 5, Conclusions and potential future research directions