2405.00317
Input gradient annealing neural network for solving low-temperature Fokker-Planck equations
Liangkai Hang, Dan Hu, Zhi-Qin John Xu
incompletemedium confidence
- Category
- Not specified
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper defines the generalized-potential Fokker–Planck residual Q(V) (its Eq. (4)) and the IGANN loss with a negative gradient penalty and an annealing schedule (its Eqs. (22)–(24)) but provides no mathematical proof that (i) constants cannot minimize the penalized loss or (ii) annealing the negative penalty leaves the β→0 limit unbiased toward solutions of the generalized-potential FPE; these are stated heuristically and supported only by experiments and remarks that β decays to recover the original loss near the end of training . By contrast, the model’s solution supplies a variational analysis under standard elliptic regularity assumptions that (a) rules out constant minimizers whenever ∇·(f−ε∇·D)≢0, and (b) proves that any accumulation point of exact global minimizers of Lβ along βk↓0 is a global minimizer of L0 (hence solves Q(V)=0), establishing the desired lack of bias in the annealed limit. The proof does impose extra hypotheses (bounded C^{1,1} domain with Dirichlet data; ρ bounded below; d≤4 for an L^4 embedding) and contains a minor sign typo in an integration-by-parts display, but the argument is otherwise logically consistent and complete for the stated setting. The paper’s claims remain empirical; the model’s proof resolves them rigorously in a classical PDE framework.
Referee report (LaTeX)
\textbf{Recommendation:} major revisions \textbf{Journal Tier:} specialist/solid \textbf{Justification:} The submission proposes a simple, effective training objective for learning generalized potentials in steady Fokker–Planck problems and demonstrates strong empirical performance, especially at low temperature. However, the central claims about the role of the negative input-gradient penalty and the unbiased nature of annealing lack theoretical backing. Adding a formal statement of conditions under which constant solutions cannot minimize the penalized loss and a theorem showing that annealed minimizers converge to minimizers of the original residual loss would bring the work to a publishable standard.