feynman-kac methods

This work addresses the challenge of effectively extending kernel-based methods—originally developed for deterministic dynamical systems—to stochastic differential equations (SDEs) for approximating eigenfunctions of the Koopman operator. By leveraging the Feynman–Kac path integral representation, the study unifies three distinct kernel constructions—variational principles, Green’s function convolutions, and resolvent operators—into a coherent framework for stochastic systems with diffusion, establishing a corresponding reproducing kernel Hilbert space (RKHS) approximation scheme. Theoretically, under uniform ellipticity, these three approaches are shown to be equivalent, revealing that diffusion enhances numerical conditioning through elliptic regularization. The analysis further provides error bounds that separate RKHS approximation error from Monte Carlo sampling error. Numerical experiments on the Ornstein–Uhlenbeck process, nonlinear SDEs, and high-dimensional systems demonstrate the method’s efficacy, showing that moderate diffusion significantly improves numerical stability.

This work proposes an efficient method for solving linear elliptic partial differential equations with constant diffusion, drift, and killing terms by integrating an enhanced Walk-on-Spheres (WoS) Monte Carlo algorithm with deep neural networks. The approach constructs unbiased estimators through explicit stochastic time sampling and employs a tailored neural network architecture to approximate both the stochastic representation of the solution and the boundary data. It represents the first integration of stochastic representations for elliptic PDEs with drift and killing terms into a deep learning framework. The study establishes uniform error bounds for the Monte Carlo estimator and proves that the solution approximation achieves polynomial complexity in both accuracy and dimensionality, thereby significantly extending the theoretical foundations and practical applicability of numerical methods for high-dimensional PDEs.

To address scalability bottlenecks in large-scale probabilistic PDE solving caused by dense covariance matrices, this paper proposes a scalable physics-informed Bayesian solver. Methodologically, it introduces the first coupling of Gaussian Markov random fields (GMRFs) with stochastic PDE (SPDE) priors, yielding an SPDE-GMRF joint prior that inherits Markovian sparsity; combined with a variational inference framework leveraging sparse linear algebra and embedded physical constraints, it explicitly models discretization error, parameter uncertainty, and observational noise. Compared to conventional dense Gaussian process approaches, the method achieves order-of-magnitude reductions in memory and runtime overhead for nonlinear PDE inverse problems, while significantly accelerating convergence. The core contribution is a unified modeling paradigm that overcomes expressivity limitations of standard covariance functions—simultaneously ensuring physical interpretability, computational scalability, and rigorous uncertainty quantification.

This work proposes a novel framework that integrates diffusion models with physics-informed sequential Monte Carlo (SMC) to generate solutions of partial differential equations (PDEs) that satisfy physical constraints. For the first time, a physics-guided mechanism based on PDE residuals is embedded directly into the stochastic sampling process of the diffusion model, enabling dynamic enforcement of multi-physics and coupled PDE constraints during generation by jointly leveraging observational data and physical laws. Experimental results across multiple benchmark PDE systems and multi-physics coupling problems demonstrate that the proposed method significantly outperforms existing state-of-the-art generative models, achieving lower numerical errors while maintaining high solution fidelity. This approach establishes a new paradigm for scalable, high-accuracy generative PDE solvers.

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 proposes the first semantically consistent sequential Monte Carlo (SMC) framework grounded in the Feynman–Kac formalism for efficient and provably correct inference in general-purpose probabilistic programs that support arbitrary measure sampling and conditional reweighting within unbounded loops. The approach employs probabilistic program graphs (PPGs) as an intermediate representation and leverages a finite-trace approximation theorem to rigorously establish the correspondence between the program’s expectation semantics and the Feynman–Kac model. Building on this foundation, the authors design a vectorized particle filtering algorithm (VPF) tailored to PPGs. Empirical evaluations demonstrate that VPF significantly outperforms existing state-of-the-art inference tools across multiple benchmarks, achieving a compelling combination of theoretical soundness, computational efficiency, and strong scalability.

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 proposes a unified variational generative modeling framework based on stochastic differential equations (SDEs) to efficiently address complex data generation tasks, including images, videos, and biomolecular structures. By incorporating both ordinary and stochastic differential equations, the authors derive the evidence lower bound (ELBO) from a variational inference perspective, systematically demonstrating that diffusion models, score matching, and flow matching are distinct parameterizations within this general framework. Through theoretical analysis grounded in the Fokker–Planck equation and empirical validation via one-dimensional density modeling experiments, the study provides clear comparisons among different parameterization strategies, confirming the proposed framework’s theoretical coherence, expressive capacity, and practical efficacy.

This work addresses the lack of uncertainty quantification in numerical solutions of the incompressible Navier–Stokes equations by proposing a Bayesian sequential inference framework. The approach models discretized dynamics as a state-space system, integrates the Feynman–Kac stochastic representation with spectral methods, and incorporates a particle learning mechanism to mitigate weight degeneracy. For the first time, this framework enables a deep integration of Navier–Stokes solvers with Bayesian inference, supporting non-Gaussian heavy-tailed error modeling, sequential data assimilation under partial observations, and scale-augmented latent variables. Demonstrated on two- and three-dimensional test cases, the method robustly propagates uncertainty through time, significantly enhances resilience to model error, and yields full posterior distributions over quantities of interest rather than point estimates.

This work addresses the absence of non-asymptotic error bounds for sequential Monte Carlo (SMC) methods employing biased proposal kernels in conditional sampling with pretrained generative models. It introduces the first non-asymptotic analysis framework that jointly controls kernel bias and particle approximation error. By extending local Doeblin conditions and Lyapunov drift arguments to conditional distributions, and integrating Feynman–Kac flow approximations with score-based diffusion models, the total error is decomposed under a forward-smoothing forgetfulness assumption into contributions from initialization, time discretization, score approximation, and finite-particle effects. This study establishes the first comprehensive non-asymptotic error bounds for conditional diffusion sampling, thereby providing theoretical guarantees for the reliability of SMC in generative modeling.