# Moment Estimate and Variational Approach for Learning Generalized Diffusion with Non-gradient Structures

arXiv: https://arxiv.org/abs/2508.01854 · HTML: https://mlsanity.com/paper/2508.01854/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 67 references that are in the map.
- First posted on arXiv 3 Aug 2025.

## Builds on (most cited first, 30 of 36)

- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
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- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Symplectic Recurrent Neural Networks](https://mlsanity.com/paper/1909.13334/) (2019)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [Machine learning structure preserving brackets for forecasting irreversible processes](https://mlsanity.com/paper/2106.12619/) (2021)
- [Nonparametric inference of interaction laws in systems of agents from trajectory data](https://mlsanity.com/paper/1812.06003/) (2018)
- [Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps](https://mlsanity.com/paper/2103.05632/) (2021)
- [Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning](https://mlsanity.com/paper/2106.09004/) (2021)
- [OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle](https://mlsanity.com/paper/2009.02327/) (2020)
- [Hamiltonian neural networks for solving equations of motion](https://mlsanity.com/paper/2001.11107/) (2020)
- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Learning interaction kernels in stochastic systems of interacting particles from multiple trajectories](https://mlsanity.com/paper/2007.15174/) (2020)
- [Learning Stochastic Dynamical System via Flow Map Operator](https://mlsanity.com/paper/2305.03874/) (2023)
- [Learning interaction kernels in mean-field equations of 1st-order systems of interacting particles](https://mlsanity.com/paper/2010.15694/) (2020)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [Efficiently Parameterized Neural Metriplectic Systems](https://mlsanity.com/paper/2405.16305/) (2024)
- [A Training-Free Conditional Diffusion Model for Learning Stochastic Dynamical Systems](https://mlsanity.com/paper/2410.03108/) (2024)
- [Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarks](https://mlsanity.com/paper/2307.11013/) (2023)
- [Constructing Custom Thermodynamics Using Deep Learning](https://mlsanity.com/paper/2308.04119/) (2023)
- [Learning Theory for Inferring Interaction Kernels in Second-Order Interacting Agent Systems](https://mlsanity.com/paper/2010.03729/) (2020)
- [Energy-Dissipative Evolutionary Deep Operator Neural Networks](https://mlsanity.com/paper/2306.06281/) (2023)
- [Learning Stochastic Behaviour from Aggregate Data](https://mlsanity.com/paper/2002.03513/) (2020)
- [A Data Driven Method for Computing Quasipotentials](https://mlsanity.com/paper/2012.09111/) (2020)
- [Coarse-Graining Hamiltonian Systems Using WSINDy](https://mlsanity.com/paper/2310.05879/) (2023)
- [Nonparametric learning of kernels in nonlocal operators](https://mlsanity.com/paper/2205.11006/) (2022)

## Built on by (most cited first, 1 of 1)

- [Moment Estimates and DeepRitz Methods on Learning Diffusion Systems with Non-gradient Drifts](https://mlsanity.com/paper/2509.10495/) (2025)

## Abstract (arXiv)

This paper proposes a data-driven learning framework for identifying governing laws of generalized diffusions with non-gradient components. By combining energy dissipation laws with a physically consistent penalty and first-moment evolution, we design a two-stage method to recover the pseudo-potential and rotation in the pointwise orthogonal decomposition of a class of non-gradient drifts in generalized diffusions. Our two-stage method is applied to complex generalized diffusion processes including dissipation-rotation dynamics, rough pseudo-potentials and noisy data. Representative numerical experiments demonstrate the effectiveness of our approach for learning physical laws in non-gradient generalized diffusions.