2202.02435
On Neural Differential Equations
Patrick Kidger
correctmedium confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s Theorem C.37 states and proves the rough-path adjoint identity for Stratonovich SDEs/RDEs under µ,σ either linear or Lip(γ+1) with γ>2, establishing existence/uniqueness of the backward rough adjoint and the identification a(t)=dL(y(T))/dy(t) almost surely; the proof proceeds by approximating with CDE adjoints, invoking the universal limit theorem, and using forward sensitivities to identify the limit . The candidate solution gives a clean pathwise proof directly at the rough level via the C^1-flow/Jacobian RDE and a rough product-rule constancy argument, yielding the same backward RDE and identification. Assumptions match the paper’s, and the reasoning aligns with standard rough-path results; thus both are correct, with different proof strategies.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The theorem and its proof are correct and well-aligned with standard rough-path theory. The approach—approximate CDE adjoints and pass to the RDE limit—carefully avoids filtration issues and delivers a clean a.s. pathwise result useful for neural SDEs. A few explicit citations to rough-flow differentiability and a short note on the rough product rule would make the presentation fully self-contained for readers less familiar with these standard facts.