analyze fokker–planck dynamics

This work addresses the numerical challenges in solving the Fokker–Planck equation under Dirac delta initial conditions and small time steps by proposing a physics-informed neural network framework based on conditional normalizing flows. The approach leverages the Chapman–Kolmogorov equation to reformulate the solution operator learning task as an approximation of transition probability density functions, using the analytical solution of a linearized stochastic differential equation as the base distribution for the normalizing flow. To mitigate instability at small time scales, a time-weighted loss function is introduced, while analytical priors are incorporated to eliminate the initial singularity. Experimental results demonstrate that the method achieves high accuracy, strong generalization, and robustness in learning solution operators across diverse complex initial conditions and small time steps.

This work addresses the unclear impact of score estimation errors on generation quality and stability in existing diffusion models. It introduces, for the first time, a stochastic partial differential equation (SPDE) framework that models score errors as stochastic sources, characterizing the evolution of the probability density field via a forward SPDE. This field-theoretic perspective enables rigorous analysis of geometric stability and displacement convexity in the generative process. The study further proposes a novel quadratic variational metric based on projections onto radial test functions, which efficiently evaluates model performance using only the first 10% of sampling trajectories. This approach not only substantially improves evaluation efficiency but also deepens the understanding of score error dynamics.

This work investigates the fundamental speed–accuracy trade-off in diffusion models, establishing—for the first time—a theoretical connection to nonequilibrium stochastic thermodynamics. Methodologically, it links the entropy production rate (a kinetic speed measure in the absence of nonconservative forces) to generation accuracy, deriving a quantitative inequality between them; it further defines an “optimal learning protocol” via the 2-Wasserstein geodesic from optimal transport theory, revealing an intrinsic Pareto frontier between speed and fidelity. The approach integrates stochastic thermodynamics, Fokker–Planck analysis, Wasserstein geometry, and numerical diffusion modeling. Experiments across diverse noise schedules and real-world image datasets validate the trade-off, quantify distortion induced by nonconservative forces, and demonstrate that the optimal protocol significantly improves sampling efficiency and reconstruction quality.

This work addresses the lack of robustness and insufficient resolution in critical regions when solving time-varying Fokker–Planck–Kolmogorov (FPK) equations for probability density evolution. We propose a Distribution-Adaptive Normalized Physics-Informed Neural Network (DA-N-PINN), which integrates soft normalization constraints with a prior-guided adaptive resampling strategy to establish global structural awareness during pretraining and dynamically concentrate sampling on high-gradient and high-probability-density regions. Furthermore, DA-N-PINN couples normalized-enhanced PINNs, weighted kernel density estimation, and a physics-constraint-driven joint optimization mechanism. Extensive validation on benchmark numerical experiments and real-world macroeconomic datasets demonstrates significant improvements in solution accuracy and computational efficiency. Notably, the method exhibits superior generalizability and robustness under sparse-data and strongly nonlinear regimes.

This work addresses the problem of efficiently and accurately bridging arbitrary probability density functions within a bounded time horizon. Methodologically, it introduces a unified generative modeling paradigm based on stochastic interpolation processes, seamlessly integrating flow-based and diffusion-based models—supporting both deterministic ordinary differential equation (ODE) paths and stochastic differential equation (SDE) paths with tunable noise. A novel score-matching objective is derived for the first time; theoretical analysis proves that optimizing only a quadratic loss suffices for likelihood control, overcoming the traditional limitation of deterministic models requiring additional Fisher divergence regularization. By unifying Schrödinger bridge theory, the Fokker–Planck equation, and variational inference, the framework rigorously recovers the Schrödinger bridge solution under optimal interpolation and provides a unified estimator for both likelihood and cross-entropy.

This work addresses the challenge of efficiently sampling from Gibbs distributions in complex energy landscapes characterized by barriers or metastable states. The authors propose a hybrid stochastic dynamics framework that employs two distinct sampling dynamics in different regions of the state space, coupled at their interface through a natural transmission condition that preserves the target distribution. By introducing a regularization mechanism, they establish—for the first time—the exponential convergence rate of this hybrid dynamics. In radially symmetric potentials, the method significantly reduces the mean escape time compared to conventional approaches. Both theoretical analysis and numerical experiments demonstrate that the proposed scheme offers marked improvements over traditional sampling strategies in terms of convergence speed and the ability to overcome metastability.

This study addresses the identifiability problem arising from the underdetermined Fokker-Planck equation when inferring dynamics from distributional snapshots. We propose a method that separates instantaneous source terms using multi-time snapshot constraints, eliminating gauge ambiguity by stacking divergence operator kernels to recover the underlying dynamics field. Theoretically, we characterize the gauge blind spots inherent in single-time constraints, prove that cross-marginal variations render hidden circulations observable, and establish elimination conditions for polynomial gauge directions. Methodologically, our approach integrates smooth test function estimation, retention of known diffusion terms, and finite-sample error separation techniques. Finally, we derive matching lower and upper bounds for conditional convergence, validate predicted gauge contraction, and elucidate the design tension between cross-slice information and covariance whitening.

This work addresses the challenge of non-physical negative probabilities arising in conventional discretizations of multidimensional Fokker–Planck equations under strong anisotropic diffusion and jump processes. To overcome this, the authors propose a novel operator-splitting “diagonal leapfrog” scheme that employs matrix exponential Krylov approximations for directional sub-operators and a factored resolvent solver to handle mixed derivative terms. By integrating a Zeno-type infinite subdivision strategy, the method guarantees non-negativity and mass conservation without requiring flux limiters. Although the resulting discrete operator is not a local M-matrix, it exhibits eventual M-matrix properties. The scheme achieves second-order accuracy in both space and time, with computational complexity O(m²N + m³), and demonstrates robustness and efficiency in high-Péclet-number regimes and scenarios featuring strong convection coupled with cross-diffusion.

This study reveals the fundamental limitations of sampling algorithms based on Wasserstein gradient flows and forward-only diffusion, which exhibit slow mixing in multimodal distributions. Methodologically, by leveraging the JKO scheme and Otto calculus combined with spectral analysis and mean first passage time theory, we demonstrate that such local gradient-driven mechanisms inherit metastability phenomena from non-equilibrium statistical physics. The core contribution lies in establishing, at a structural level, that multimodal transport bottlenecks cannot be overcome by local mechanisms. Furthermore, we prove that introducing intermediate distributions or log-linear annealing strategies fails to eliminate the exponential scaling of total transport time. These findings delineate fundamental boundaries for standard sampling paradigms and advocate for the exploration of novel non-local sampling mechanisms.

This work addresses the challenge of inferring unknown population dynamics solely from snapshots of time-series probability distributions, without access to individual trajectories or prescribed dynamical equations. To this end, it introduces a novel paradigm that decomposes the dynamics into a latent Ornstein–Uhlenbeck stochastic process and a geometric transport map. The former yields an analytically tractable Fokker–Planck equation, while the latter employs monotone neural networks to implement a Knothe–Rosenblatt rearrangement that captures distributional deformations. A deformation energy regularizer grounded in hyperelasticity theory is incorporated to enhance solution uniqueness and physical interpretability. Experiments demonstrate that the method accurately reconstructs complex probabilistic dynamics in nonlinear, multimodal distribution evolution tasks, while maintaining a compact and analytically manageable latent representation.