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arxiv: 2606.24999 · v1 · pith:WMCFTZJZnew · submitted 2026-06-23 · 💻 cs.LG · math.OC

A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients

Yanwei Jia , Du Ouyang , Huy\^en Pham , Xun Yu Zhou This is my paper

Pith reviewed 2026-06-26 00:08 UTC · model grok-4.3

classification 💻 cs.LG math.OC
keywords zeroth-order derivativesdeep learningparabolic PDEsunknown coefficientsMonte Carlo trajectoriesmodel-freeSobolev spacestatistical learning
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The pith

Zeroth-order derivative estimators from perturbed Monte Carlo trajectories enable model-free learning of solutions and derivatives for high-dimensional nonlinear parabolic PDEs using only simulators.

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

The paper establishes a representing-then-learning method that solves fully nonlinear parabolic PDEs with unknown coefficients by generating data through two types of simulators and representing derivatives via zeroth-order estimators. This approach avoids automatic differentiation and explicit knowledge of the data-generating dynamics, instead using only pointwise function evaluations from simulations to create targets for neural networks approximating the solution, gradient, and Hessian. A statistical analysis provides non-asymptotic error bounds that decompose the total error and characterize sample complexity in weighted Sobolev space up to second-order derivatives, under the assumption of a contractive PDE operator. This matters for applications like continuous-time reinforcement learning and scientific machine learning where environments are black-box.

Core claim

By introducing simulators as data-generating mechanisms and deriving zeroth-order derivative estimators from perturbed Monte Carlo trajectories, the method learns solutions and their derivatives for fully nonlinear parabolic PDEs under settings where PDE operators are accessible only through simulations and pointwise evaluations, with a bias-variance analysis for the estimators and non-asymptotic error bounds that separate discretization, approximation, statistical, and ZOD bias contributions while giving sample complexity in Sobolev space.

What carries the argument

Zeroth-order derivative (ZOD) estimators derived from perturbed Monte Carlo trajectories, which generate targets for gradient and Hessian networks using only function evaluations in a fully model-free manner.

If this is right

  • The method generates targets for the gradient and Hessian networks using only function evaluations from simulators.
  • A bias-variance tradeoff analysis applies to the ZOD estimators.
  • The total error decomposes into discretization error, approximation error, statistical error, and ZOD bias under the contraction assumption.
  • Sample complexity is characterized for the learned representations in weighted Sobolev space up to second-order derivatives.
  • Numerical tests demonstrate competitive performance in moderate and high dimensions.

Where Pith is reading between the lines

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

  • The simulator-based framework could extend to other black-box settings such as continuous-time reinforcement learning without requiring known dynamics.
  • Replacing automatic differentiation with ZOD estimators may reduce instability in high-dimensional derivative computations for PDE solvers.
  • The error decomposition suggests targeted improvements by reducing ZOD bias through adjusted perturbation sizes in the Monte Carlo trajectories.

Load-bearing premise

The underlying PDE operator satisfies a standard contraction property.

What would settle it

Numerical experiments on a contractive PDE where the observed total error exceeds the bound by more than the sum of the four decomposed terms, or where performance degrades sharply when the contraction property is removed.

Figures

Figures reproduced from arXiv: 2606.24999 by Du Ouyang, Huy\^en Pham, Xun Yu Zhou, Yanwei Jia.

Figure 1
Figure 1. Figure 1: The learned value function, gradient, and Hessian for example (6.1). NN-autodiff learns the value function by the least-squares method and obtains derivatives by automatic differentiation. ZOD-m trains value, gradient, and Hessian networks directly using multi-point ZOD estimators. and the nonlinear source term is f(t, x, ∇u) = x · ∇u(t, x) − 1 2 |∇u(t, x)| 2 − d. Then we can find the exact solution u ∗ of… view at source ↗
Figure 2
Figure 2. Figure 2: rRMSE of the learned value, gradient, and Hessian over iterations for the PDE (6.2) with dimension d = 20. ZOD-m and DPI use the same network (6.5) for value function [PITH_FULL_IMAGE:figures/full_fig_p032_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: compares the time-matched ZOD run with the 40-iteration DPI run. The rapid decay of the ZOD errors in the first few value (Picard) iterations is consistent with the contraction estimate in Theorem 6. After a small number of Picard iterations, the curves approach stable accuracy levels. This behavior agrees with the structure of (5.3): once the one-step learning error has been reduced to a fixed floor, the … view at source ↗
Figure 4
Figure 4. Figure 4: Ablations for the fully nonlinear PDE (6.6). All curves report the mean rRMSE over 3 seeds, and shaded regions indicate one empirical standard deviation. further inspires us to formulate learning the gradient and Hessian also as policy evaluation problems.14 The approach developed in this paper hints on an important implication in the study of data-driven methodologies in machine learning: the design of al… view at source ↗
read the original abstract

High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging. Existing deep learning solvers often rely on repeated automatic differentiation to evaluate differential operators, which can cause instability and amplify derivative errors in high dimensions, while probabilistic methods based on stochastic representations require explicit knowledge of the data-generating dynamics and therefore do not apply to black-box environments. We introduce two types of simulators as data-generating mechanisms, and take a ``representing-then-learning" approach that learns the solutions and their derivatives under settings where the underlying PDE operators are accessible only through simulations and pointwise evaluations. Our representation of derivatives relies on the zeroth-order derivative (ZOD) estimators derived from perturbed Monte Carlo trajectories. This fully model-free approach generates targets for the gradient and Hessian networks using only function evaluations. We provide a statistical learning analysis of the proposed approach, including a bias--variance tradeoff for ZODs. Assuming a standard contraction property of the underlying operator, we establish a non-asymptotic error bound that decomposes the total error into discretization error, approximation error, statistical error, and ZOD bias. Crucially, we derive the sample complexity of the learned representations in (weighted) Sobolev space, characterizing the error up to second-order derivatives. Numerical experiments illustrate the competitive performance of the method in moderate and high dimensions.

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

2 major / 1 minor

Summary. The paper proposes a zeroth-order deep learning method for high-dimensional fully nonlinear parabolic PDEs with unknown coefficients, using two simulator types as data-generating mechanisms and a representing-then-learning approach. Derivatives are represented via ZOD estimators from perturbed Monte Carlo trajectories, enabling model-free learning from function evaluations only. The authors provide a statistical learning analysis with bias-variance tradeoff for ZODs and, assuming a standard contraction property of the operator, derive a non-asymptotic error bound decomposing total error into discretization, approximation, statistical, and ZOD bias terms, along with sample complexity results in weighted Sobolev space up to second-order derivatives. Numerical experiments demonstrate competitive performance in moderate and high dimensions.

Significance. If the contraction assumption holds and the bounds are rigorously derived without circularity, the work would provide a notable contribution to model-free solvers for black-box PDEs in scientific machine learning and continuous-time RL, addressing instabilities from automatic differentiation and the limitations of probabilistic methods requiring explicit dynamics. The explicit decomposition of errors and Sobolev-space sample complexity characterization would be strengths if supported by complete derivations.

major comments (2)
  1. [statistical learning analysis section / non-asymptotic error bound] The non-asymptotic error bound (stated in the abstract and established in the statistical learning analysis section) decomposes total error into discretization, approximation, statistical, and ZOD bias terms only under the assumption of a standard contraction property of the underlying operator. For fully nonlinear parabolic PDEs where the operator is accessible solely through simulators and pointwise evaluations with unknown coefficients, this property is not explicitly derived from the data-generating mechanism or verified, rendering the bound and the associated sample complexity in weighted Sobolev space conditional on an uncheckable hypothesis in the claimed black-box regime.
  2. [statistical learning analysis section] The ZOD bias analysis and bias-variance tradeoff (abstract and statistical learning analysis) are presented as part of the error decomposition, but without full derivations or explicit verification of how the contraction property interacts with the perturbed Monte Carlo trajectories in the black-box setting, the support for the central non-asymptotic claims cannot be fully assessed from the provided analysis.
minor comments (1)
  1. Notation for the two simulator types and the precise definition of the weighted Sobolev space should be clarified with explicit references to prior work on ZOD estimators to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. Below we respond point by point to the major comments on the statistical learning analysis and error bounds.

read point-by-point responses

  1. Referee: [statistical learning analysis section / non-asymptotic error bound] The non-asymptotic error bound (stated in the abstract and established in the statistical learning analysis section) decomposes total error into discretization, approximation, statistical, and ZOD bias terms only under the assumption of a standard contraction property of the underlying operator. For fully nonlinear parabolic PDEs where the operator is accessible solely through simulators and pointwise evaluations with unknown coefficients, this property is not explicitly derived from the data-generating mechanism or verified, rendering the bound and the associated sample complexity in weighted Sobolev space conditional on an uncheckable hypothesis in the claimed black-box regime.

    Authors: We agree that the non-asymptotic bound is derived under the standard contraction assumption on the operator. This assumption is ubiquitous in the analysis of fully nonlinear PDEs and continuous-time RL (ensuring uniqueness and convergence of the fixed-point iteration) and is stated explicitly in the manuscript. In the black-box regime the simulators define the operator only implicitly via pointwise evaluations, so an explicit derivation from the data-generating mechanism would require additional structure on the unknown coefficients, defeating the model-free objective. The assumption remains checkable numerically on concrete instances. We will add a clarifying remark on its role and typical verification in the revised version. revision: partial

  2. Referee: [statistical learning analysis section] The ZOD bias analysis and bias-variance tradeoff (abstract and statistical learning analysis) are presented as part of the error decomposition, but without full derivations or explicit verification of how the contraction property interacts with the perturbed Monte Carlo trajectories in the black-box setting, the support for the central non-asymptotic claims cannot be fully assessed from the provided analysis.

    Authors: The complete derivations of the ZOD bias, bias-variance tradeoff, and the full non-asymptotic error bound (including the manner in which the contraction controls error propagation through the operator applied to perturbed trajectories) appear in Section 4 and the appendix. Nevertheless, we accept that the interaction between the black-box simulators and the contraction could be spelled out more explicitly. We will revise the statistical learning section to include additional intermediate steps clarifying this interaction while preserving the model-free character of the approach. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation conditional on external assumption

full rationale

The paper establishes its non-asymptotic error bound only after explicitly assuming a standard contraction property of the underlying operator, which is treated as an independent precondition rather than derived from the ZOD estimators, simulators, or any fitted quantities. No equations or steps in the provided text reduce the sample complexity result or the representing-then-learning approach to inputs by construction, and there are no self-citations invoked as load-bearing uniqueness theorems. The statistical analysis (bias-variance tradeoff, discretization/approximation/statistical/ZOD bias decomposition) therefore remains self-contained against the stated external benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central error analysis rests on one domain assumption; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption standard contraction property of the underlying operator
    Invoked explicitly to derive the non-asymptotic error bound and sample complexity.

pith-pipeline@v0.9.1-grok · 5795 in / 1216 out tokens · 18566 ms · 2026-06-26T00:08:32.695815+00:00 · methodology

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Reference graph

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