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arxiv: 2605.14643 · v1 · pith:GCTLFKKUnew · submitted 2026-05-14 · 💻 cs.LG · cs.NA· math.NA· math.OC

Unbiased and Second-Order-Free Training for High-Dimensional PDEs

Jaemin Seo , Surin Lee , Jae Yong Lee This is my paper

Pith reviewed 2026-06-30 21:26 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NAmath.OC
keywords backward stochastic differential equationshigh-dimensional PDEsunbiased trainingEuler-Maruyama discretizationdeep learning for PDEssecond-order-free methodsphysics-informed neural networks
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The pith

A new training framework removes Euler-Maruyama bias from BSDE PDE solvers without second-order derivatives or added overhead.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

BSDE methods solve high-dimensional PDEs by using probabilistic representations that avoid the curse of dimensionality and often skip explicit second derivatives. The standard Euler-Maruyama time discretization creates an intrinsic bias inside the training loss. Higher-order schemes can cancel the bias but bring back second-order spatial derivatives and extra computation. The paper analyzes the source of the EM bias and introduces a modified loss that removes it while staying second-order free.

Core claim

The paper shows through analysis that the Euler-Maruyama discretization induces a specific bias in BSDE training losses. It proposes an unbiased training framework that corrects this bias directly in the loss function, preserving the second-order-free property and computational advantages of BSDE methods for high-dimensional PDEs.

What carries the argument

The unbiased loss function obtained from the EM bias analysis, which adjusts the training objective to cancel the discretization error without Hessian evaluations.

If this is right

  • BSDE-based solvers achieve unbiased training at the cost level of standard Euler-Maruyama schemes.
  • High-dimensional PDE accuracy improves without raising the order of spatial derivatives required.
  • The probabilistic representation advantages of BSDE methods remain intact.
  • Computational cost stays comparable to first-order methods even after bias removal.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The bias-correction approach could apply to other time discretizations used in stochastic differential equation solvers.
  • Reduced bias accumulation might improve accuracy for PDEs integrated over long time horizons.
  • Similar loss adjustments could be tested in related deep-learning PDE methods that rely on probabilistic representations.

Load-bearing premise

The bias correction can be applied exactly in the loss during training without creating new approximation errors or needing extra derivative computations.

What would settle it

Compare solution errors of the proposed method against standard EM and Heun schemes on a high-dimensional PDE with known analytic solution and check whether bias-induced error terms disappear at the predicted rate.

Figures

Figures reproduced from arXiv: 2605.14643 by Jaemin Seo, Jae Yong Lee, Surin Lee.

Figure 1
Figure 1. Figure 1: Illustration of the one-step loss computation for Un-EM-BSDE with M1 = M2 = 2. The many gray curves visualize the randomness of solution trajectories (possible SDE sample paths). The dashed curves denote a ground-truth trajectory, whereas the solid curves indicate the one-step predicted trajectory. The errors from the two independently formed groups are multiplied to produce an unbiased estimator. are assi… view at source ↗
Figure 2
Figure 2. Figure 2: Time-resolved relative error on the BSB benchmark: relative error evaluated at each PDE time t ∈ T using the same models as in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Relative error over PDE time (left) and RL2 over training runtime (right) for Shotgun and its debiased variant. Concretely, applying the same randomized construction to Shotgun yields the following debiased one-step loss ℓ M1,M2 USG (θ, n, x) = E[(ShotM1 [errSG n (x; uθ)]) × (ShotM2 [errSG n (x; uθ)])]. (21) We then evaluate this plug-in variant on the BSB hard￾constraint benchmark. Algorithmically, we ret… view at source ↗
Figure 5
Figure 5. Figure 5: HJB(d = 100) path prediction with error band in the soft-constraint setting. We therefore focus on balanced configurations and report results for M1 = M2 ∈ {1, 2, 5, 10, 20, 50} in [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: HJB(d = 100) path prediction with error band in the hard-constraint setting [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: BSB(d = 100) path prediction with error band in the soft-constraint setting. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: BSB(d = 100) path prediction with error band in the hard-constraint setting. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
read the original abstract

Deep learning methods based on backward stochastic differential equations (BSDEs) have emerged as competitive alternatives to physics-informed neural networks (PINNs) for solving high-dimensional partial differential equations (PDEs). By leveraging probabilistic representations, BSDE approaches can avoid the curse of dimensionality and often admit second-order-free training objectives that do not require explicit Hessian evaluations. It has recently been established that the commonly used Euler-Maruyama (EM) time discretization induces an intrinsic bias in BSDE training losses. While high-order schemes such as Heun can fully eliminate this bias, such schemes re-introduce second-order spatial derivatives and incur substantial computational overhead. In this work, we provide a principled analysis of EM-induced loss bias and propose an unbiased, second-order-free training framework that preserves the computational advantages of BSDE methods. Our code is available at https://github.com/seojaemin22/Un-EM-BSDE.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript analyzes the bias induced by Euler-Maruyama (EM) time discretization in the loss functions of backward stochastic differential equation (BSDE) methods for high-dimensional PDEs. It proposes a new unbiased training framework that removes this bias while remaining second-order-free, thereby preserving the computational advantages of standard BSDE approaches over PINNs and higher-order schemes such as Heun. The work includes open-source code.

Significance. If the analysis and construction hold, the result would be significant for scalable, high-dimensional PDE solvers: it directly addresses a recently identified source of bias in BSDE training without sacrificing the second-order-free property or incurring the overhead of high-order time stepping. The public code repository supports reproducibility.

minor comments (1)
  1. The abstract and introduction would benefit from a brief, explicit statement of the precise form of the proposed unbiased loss (e.g., the correction term or modified expectation) to allow readers to assess the construction before reaching the technical sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and for recommending minor revision. The referee correctly identifies the core contribution: an analysis of Euler-Maruyama bias in BSDE losses together with an unbiased yet second-order-free training framework that retains the computational advantages of standard BSDE methods.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a principled analysis of EM-induced bias in BSDE training losses followed by a construction for an unbiased second-order-free objective. No load-bearing step reduces by definition to its own inputs, no fitted parameter is relabeled as a prediction, and no uniqueness theorem or ansatz is imported solely via self-citation. The central claim rests on an independent bias analysis whose validity is not presupposed by the proposed framework itself; the derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.1-grok · 5689 in / 984 out tokens · 24251 ms · 2026-06-30T21:26:25.393597+00:00 · methodology

discussion (0)

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Reference graph

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