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arxiv: 2604.20188 · v1 · submitted 2026-04-22 · 💻 cs.LG · math.DS

Structure-Aware Variational Learning of a Class of Generalized Diffusions

Yubin Lu , Xiaofan Li , Chun Liu , Qi Tang , Yiwei Wang This is my paper

Pith reviewed 2026-05-10 00:10 UTC · model grok-4.3

classification 💻 cs.LG math.DS
keywords variational learningenergy-based modelsgeneralized diffusionsFokker-Planck equationDe Giorgi dissipationpotential function inferencestochastic processesstructure-aware learning
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The pith

An energy-based variational loss from the Fokker-Planck energy-dissipation law infers unknown potentials in generalized diffusions without direct PDE enforcement.

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

The paper develops a learning method for the potential energy driving stochastic diffusion processes when data is incomplete and noisy. Classical methods that regress equations or velocities directly often falter under these conditions. By grounding the loss in the energy-dissipation structure rather than the PDE itself, the approach preserves the system's variational properties. Experiments across one to three dimensions show the resulting loss remains stable as observation times vary, noise increases, or training data becomes sparser or less diverse.

Core claim

Starting from the energy-dissipation law of the Fokker-Planck equation, the authors construct loss functions using the De Giorgi dissipation functional. These losses couple the free energy and the dissipation mechanism without explicitly enforcing the governing PDE, allowing structure-aware inference of the unknown potential function in generalized diffusion processes.

What carries the argument

The De Giorgi dissipation functional built from the energy-dissipation law associated with the Fokker-Planck equation, which acts as the variational loss that couples free energy and dissipation for learning the potential.

If this is right

  • The method recovers potentials accurately even with limited or noisy trajectory data.
  • Robustness improves across varying observation times and noise levels in 1D, 2D, and 3D settings.
  • The variational structure is preserved, enabling consistent coupling of energy and dissipation without PDE enforcement.
  • Performance holds with diverse and reduced amounts of training data.

Where Pith is reading between the lines

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

  • The approach could extend to learning in other variational systems where energy-dissipation laws are known but full PDE solutions are intractable.
  • If the potential is learned this way, downstream simulations of the diffusion process should match observed statistics more reliably than direct regression methods.
  • Testing on real experimental data from chemistry or biology would reveal whether the robustness observed in numerics translates to physical systems.

Load-bearing premise

The energy-dissipation law from the Fokker-Planck equation and the De Giorgi functional can be turned into a loss that recovers the true potential without needing to enforce the PDE directly or have complete observations.

What would settle it

Generate synthetic trajectories from a known potential in a generalized diffusion, add high noise and remove some observations, then check if minimizing the proposed loss recovers a potential whose simulated trajectories match the original statistics; failure to do so would falsify the robustness claim.

Figures

Figures reproduced from arXiv: 2604.20188 by Chun Liu, Qi Tang, Xiaofan Li, Yiwei Wang, Yubin Lu.

Figure 1
Figure 1. Figure 1: Learning the potential ψ(x) = 1 4 x 4 − 1 2 x 2 using the loss function (24) for different values of the weighting parameter α. As shown in [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Learning the quadruple-well potential ψ(x, y) = 1 4 x 4 − 1 2 x 2 + 1 4 y 4 − 1 2 y 2 using training data collected over different time intervals [Tb, Te]. Panel (a) corresponds to α = 1, and panel (b) corresponds to α = 0.5. In each panel, the subplots are arranged from left to right and top to bottom, showing the ground-truth potential and the learned potentials obtained using data from the time interval… view at source ↗
Figure 3
Figure 3. Figure 3: Learning the quadruple-well potential ψ(x, y) = 1 4 x 4− 1 2 x 2+ 1 4 y 4− 1 2 y 2 . Panel (a) shows the impact of the number of initial distributions Q for α = 1, while panel (b) illustrates the robustness of the learning results with respect to different noise levels for α = 0.5. Impact of Observational Noise In this experiment, we examine the robustness of the De Giorgi dissipation functional–based loss… view at source ↗
Figure 4
Figure 4. Figure 4: compares the learned potential using the loss function (24) for different values of the weighting parameter α = 0.25, 0.5, 0.75. Both α = 0.5 and α = 0.25 are better than α = 0.75. Moreover, when examining the contour details, α = 0.5 also shows some improvement over α = 0.25. Overall, α = 0.5 appears to be the optimal choice. −  [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between the learned potential and the true potential given by Eq. (26). −  x −  [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Evolution of particle trajectories within the learned two-dimensional potential. The red solid points represent the particle positions at four distinct time points, simulated using the stochastic differential equation given in Eq. (1), with the dynamics driven by the learned potential. where (q1, q2, q3) = (x, y, z). The parameters are chosen as (c1, c2, c3) = (0.2, 0.3, 0.3), (A1, A2, A3) = (−2.0, −2.0, −… view at source ↗
Figure 7
Figure 7. Figure 7: Learning the three-dimensional potential (27) with observational noise level σ = 0.2. The three columns correspond to the density plot of the potential function in the z = 0, y = 0, and x = 0 coordinate planes respectively. The first row shows the ground-truth potential, and the second row shows the learned potential. 0     t 0  0 0 00 t [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Learning the three-dimensional potential (27) with observational noise level σ = 0.2. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: presents the learning results obtained using the loss functions (24) with α = 0.5 and (25). It is evident that, under the presence of the environmental velocity perturbation (28), the energy-based loss (24) yields significantly more accurate reconstructions of the target potential than the PDE-based loss (25). This behavior can be explained by the structural difference between the two approaches. The PDE-b… view at source ↗
read the original abstract

Learning the underlying potential energy of stochastic gradient systems from partial and noisy observations is a fundamental problem arising in physics, chemistry, and data-driven modeling. Classical approaches often rely on direct regression of governing equations or velocity fields, which can be sensitive to noise and external perturbations and may fail when observations are incomplete. In this work, we propose a structure-aware, energy-based learning framework for inferring unknown potential functions in generalized diffusion processes, grounded in the energetic variational approach. Starting from the energy-dissipation law associated with the Fokker-Planck equation, we construct loss functions based on the De Giorgi dissipation functional, which consistently couple the free energy and the dissipation mechanism of the system. This formulation avoids explicit enforcement of the governing partial differential equation and preserves the underlying variational structure of the dynamics. Through numerical experiments in one, two, and three dimensions, we demonstrate that the proposed energy-based loss exhibits enhanced robustness with respect to observation time, noise level, and the diversity and amount of available training data. These results highlight the effectiveness of energy-dissipation principles as a reliable foundation for learning stochastic diffusion dynamics from data.

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 / 2 minor

Summary. The paper proposes a structure-aware variational learning framework for inferring unknown potential functions in generalized diffusion processes from partial and noisy observations. Grounded in the energetic variational approach, it starts from the energy-dissipation law of the Fokker-Planck equation and constructs loss functions via the De Giorgi dissipation functional to couple free energy and dissipation without explicit PDE enforcement. Numerical experiments in one, two, and three dimensions are used to claim enhanced robustness with respect to observation time, noise level, and training data diversity and volume.

Significance. If the robustness claims are substantiated with proper quantitative validation, the work would provide a principled, structure-preserving alternative to direct regression methods for learning stochastic dynamics, leveraging established energy-dissipation principles and De Giorgi functionals. This could be valuable for physics-informed machine learning in inverse problems involving diffusions. The approach avoids explicit PDE enforcement while retaining variational structure, which is a positive aspect, but the current lack of metrics limits assessment of its practical advantage.

major comments (2)
  1. [Abstract and numerical experiments] Abstract and numerical experiments section: The claim that 'numerical experiments in one, two, and three dimensions... demonstrate that the proposed energy-based loss exhibits enhanced robustness' is load-bearing for the central contribution, yet the text supplies no quantitative metrics, baseline comparisons, error bars, data-generation protocols, or exclusion rules. This prevents independent verification of the data-to-claim link.
  2. [Method (energy-dissipation construction)] Method construction (energy-dissipation law and De Giorgi functional): The loss may admit multiple potentials consistent with partial noisy observations without strict convexity or injectivity guarantees; different potentials could produce similar energy-dissipation balances on observed trajectories. No analysis of loss-landscape flatness, identifiability, or controlled recovery error under perturbations of the true potential is provided, which is required to support the inference claim.
minor comments (2)
  1. [Abstract] The abstract would be clearer if it briefly indicated the explicit form of the constructed loss function or referenced the key variational equations.
  2. [References] Ensure citations to prior literature on energetic variational approaches and applications of De Giorgi functionals to learning problems are complete and up-to-date.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review of our manuscript. We address each major comment point by point below, indicating the revisions we will undertake to strengthen the presentation and support for our claims.

read point-by-point responses

  1. Referee: [Abstract and numerical experiments] Abstract and numerical experiments section: The claim that 'numerical experiments in one, two, and three dimensions... demonstrate that the proposed energy-based loss exhibits enhanced robustness' is load-bearing for the central contribution, yet the text supplies no quantitative metrics, baseline comparisons, error bars, data-generation protocols, or exclusion rules. This prevents independent verification of the data-to-claim link.

    Authors: We agree that the numerical experiments section requires more rigorous quantitative support to substantiate the robustness claims. In the revised manuscript, we will expand this section to include explicit quantitative metrics (e.g., mean squared error in recovered potentials with standard deviations over repeated trials), direct comparisons against baselines such as least-squares drift regression and physics-informed neural network methods, full descriptions of data-generation protocols (including ranges for observation times, noise variances, and training set sizes), and any data exclusion or preprocessing rules applied. These changes will enable independent verification and clearer evaluation of the method's advantages. revision: yes

  2. Referee: [Method (energy-dissipation construction)] Method construction (energy-dissipation law and De Giorgi functional): The loss may admit multiple potentials consistent with partial noisy observations without strict convexity or injectivity guarantees; different potentials could produce similar energy-dissipation balances on observed trajectories. No analysis of loss-landscape flatness, identifiability, or controlled recovery error under perturbations of the true potential is provided, which is required to support the inference claim.

    Authors: We acknowledge that the inverse problem is inherently ill-posed under partial and noisy observations, and that the loss may not guarantee unique recovery without additional assumptions. The De Giorgi-based construction is motivated by preserving the energy-dissipation structure rather than enforcing uniqueness a priori. In the revision, we will add a dedicated discussion of identifiability conditions (e.g., sufficient state-space coverage) together with new numerical experiments that report controlled recovery errors under perturbations of the ground-truth potential and basic diagnostics of loss-landscape behavior. While a full theoretical proof of strict convexity lies beyond the current scope, these empirical and conditional analyses will better support the inference claims. revision: partial

Circularity Check

0 steps flagged

No significant circularity; construction applies established variational principles

full rationale

The paper starts from the established energy-dissipation law of the Fokker-Planck equation and applies the De Giorgi dissipation functional to build an energy-based loss. This is a direct application of known structure from the energetic variational approach rather than any self-definitional loop, fitted input renamed as prediction, or load-bearing self-citation chain. No equations reduce the claimed robustness to tautology or prior author-specific uniqueness results. The numerical experiments in 1D/2D/3D serve as external validation of the loss properties and do not close a circular derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the standard energetic variational approach and the energy-dissipation law of the Fokker-Planck equation; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Generalized diffusion processes obey the energy-dissipation law associated with the Fokker-Planck equation.
    The method starts from this law to construct the loss functions, as stated in the abstract.

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