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arxiv: 2411.13443 · v4 · submitted 2024-11-20 · 🧮 math.NA · cs.NA· math.OC· stat.ML

Nonlinear Assimilation via Score-based Sequential Langevin Sampling

Zhao Ding , Chenguang Duan , Yuling Jiao , Jerry Zhijian Yang , Cheng Yuan , Pingwen Zhang This is my paper

Pith reviewed 2026-05-23 16:52 UTC · model grok-4.3

classification 🧮 math.NA cs.NAmath.OCstat.ML
keywords nonlinear data assimilationscore-based Langevin samplingBayesian filteringasymptotic stabilitytotal variation convergenceannealing strategyerror boundsuncertainty quantification
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The pith

Score-based sequential Langevin sampling establishes asymptotic stability for nonlinear data assimilation by bounding error accumulation.

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

The paper introduces score-based sequential Langevin sampling (SSLS) as a recursive Bayesian filter that splits assimilation into a prediction step using dynamic models and an update step that draws from the posterior via score-based Langevin Monte Carlo. An annealing schedule is added to the updates to handle posteriors that are far from log-concave. The authors prove that the method converges in total variation distance and that the resulting error bounds remain stable, so that local sampling inaccuracies do not grow without limit across successive assimilation cycles. Numerical tests on high-dimensional, strongly nonlinear problems with sparse data show that the procedure also produces usable uncertainty estimates.

Core claim

SSLS decomposes nonlinear assimilation into alternating prediction and update phases, employs score-based Langevin dynamics with annealing during the update phase, and supplies explicit total-variation error bounds that establish asymptotic stability: local posterior-sampling errors remain controlled and do not accumulate indefinitely over time.

What carries the argument

Score-based sequential Langevin sampling (SSLS) with integrated annealing inside the recursive Bayesian update step.

If this is right

  • The derived total-variation bounds give explicit dependence of the error on the number of Langevin steps and the annealing schedule.
  • Local sampling errors remain bounded across arbitrary numbers of assimilation cycles.
  • The method supplies calibrated uncertainty estimates for the state trajectory in high-dimensional nonlinear settings.
  • Performance holds under sparse observations and strong nonlinearity.

Where Pith is reading between the lines

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

  • The stability result could be checked by monitoring total-variation distance on a controlled linear-Gaussian problem where the exact posterior is known.
  • If annealing is the key enabler, replacing it with other tempering schemes might preserve the same error bounds.
  • The recursive structure suggests direct extension to online filtering where new observations arrive continuously.

Load-bearing premise

The annealing schedule is sufficient to make score-based Langevin sampling reliable even for highly non-log-concave posteriors.

What would settle it

A sequence of assimilation steps long enough that the total-variation distance between the SSLS output and the true filtering distribution grows unbounded.

Figures

Figures reproduced from arXiv: 2411.13443 by Chenguang Duan, Cheng Yuan, Jerry Zhijian Yang, Pingwen Zhang, Yuling Jiao, Zhao Ding.

Figure 1
Figure 1. Figure 1: An illustrative schematic of the state-space model. The latent states (Xk)k∈N are unobservable and evolves according to known transition densities (ρk)k∈N, which are specified by a dynamics model (2.1). The observations (Yk)k∈N are linked with states by a known likelihood gk characterized by the measurement model (2.2). direct access to statistical inference through the computation of essential measures su… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic representation of score-based sequential Langevin sam￾pling. (Left) The prediction step involves sampling from the approximated prediction distribution and estimating the prediction score. (Right) The poste￾rior score is then obtained by combining the prediction score with the gradient of the log-likelihood. The update step samples from the posterior distribution using ALMC. Combining these two p… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic comparison of vanilla and annealed Langevin algorithms. (Top) The vanilla Langevin algorithm samples from the target posterior distribution, using the prediction distribution as initialization. (Bottom) The annealed Langevin algorithm employs a sequence of interpolations that smoothly transition from the prediction distribution to the target posterior distribution. Since the Langevin diffusion (2… view at source ↗
Figure 4
Figure 4. Figure 4: Results of assimilation for Langevin diffusion with a double-well potential (4.2) with a linear measurement model (4.3). The ensemble mean of SSLS, APF, and EnKF at each time steps are shown in the figure. Svensson, 2023, Chapter 11.6). This phenomenon, known as particle degeneracy, persists even though APF offers some improvement over the standard PF. In contrast, SSLS and EnKF avoid reliance on particle … view at source ↗
Figure 5
Figure 5. Figure 5: Results of assimilation for Langevin diffusion with a double-well potential (4.2) with a nonlinear measurement model (4.4). The ensemble mean of SSLS, APF, and EnKF at each time steps are shown in the figure. where u represents the velocity field, Re is the Reynolds number, ρ denotes the fluid density, p is the pressure field, and F is the external forcing. This example uses the periodic boundary condition… view at source ↗
Figure 6
Figure 6. Figure 6: Results of assimilation for Kolmogorov flow (4.5) with different measurement models. Each column corresponds to distinct time steps (states are plotted for every 10 time steps). The first row displays the reference state. (i) Super-resolution: The 2nd and 3rd rows display the noisy observations with 8x average pooling and the corresponding SSLS estimations, respectively. (ii) Sparse reconstruction: The 4th… view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of results of SSLS and methods without prediction score. From top to bottom: the reference state, observations, estimations of the ensemble MLE, estimations of Langevin sampling without the prediction score, and estimations of SSLS. Here the noise level is set as σobs = 0.3 and let 90% points be randomly masked. (method (i)) or remain unchanged (method (ii)). This spatially inconsistent updating… view at source ↗
Figure 8
Figure 8. Figure 8: Quantify the uncertainty associated with states estimated by SSLS. From top to bottom: the reference states, observations (95% random mask), the SSLS assimilated states, point-wise error (in absolute value) and standard deviation. The noise level is set as σobs = 0.4. Standard deviation and uncertainty. A notable advantage of SSLS is its ability to generate multiple ensemble samples from the posterior dist… view at source ↗
Figure 9
Figure 9. Figure 9: The correlation between the standard deviation and estimation error of the SSLS. The standard deviation and error are down-sampled by max pooling for clearer visualization. From left to right: the results at three equally separated time points of the assimilation process. The last two rows of [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Posterior distributions estimated by SSLS in a linear Gaussian state￾space model (J.1). (a) The top row shows the histogram of the SSLS ensemble with an exact initial prior distribution. (b) The bottom row demonstrates the histogram of the SSLS ensemble with an inexact initial prior distribution. Recall that Theorem 3.4 provides an error bound that increases with the number of time step. Nonetheless, this… view at source ↗
Figure 11
Figure 11. Figure 11: Evolution of the reference states, the SSLS ensemble mean, and the APF ensemble mean for Lorenz-96 (J.2) [PITH_FULL_IMAGE:figures/full_fig_p055_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Performance metrics of SSLS and APF for Lorenz-96 (J.2). For each ensemble size, metrics are averaged over elements of the estimated states and time steps. Experimental results [PITH_FULL_IMAGE:figures/full_fig_p056_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Evolution of x10 of the true states, the SSLS ensemble and the APF ensemble on REFERENCE LORENZ [PITH_FULL_IMAGE:figures/full_fig_p057_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Trajectory of the true states, the SSLS estimation and the APF estimation for a Lorenz 96 system. The trajectory is visualized in the x1-x20 space. 0 10 20 30 40 50 Time step 10 1 10 0 RMSE AvgPool scale 2 4 8 [PITH_FULL_IMAGE:figures/full_fig_p058_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: RMSE of SSLS assimilated states at different average pooling scale. Time 0 corresponds to RMSE from the expectation of the prior distribution, when the assimilation has not taken place. The three lines share the same starting RMSE as they share the same guess on the prior distribution. the true state. In [PITH_FULL_IMAGE:figures/full_fig_p058_15.png] view at source ↗
read the original abstract

This paper introduces score-based sequential Langevin sampling (SSLS), a novel approach to nonlinear data assimilation within a recursive Bayesian filtering framework. The proposed method decomposes the assimilation process into alternating prediction and update steps, using dynamic models for state prediction and incorporating observational data via score-based Langevin Monte Carlo during the updates. To overcome inherent challenges in highly non-log-concave posterior sampling, we integrate an annealing strategy into the update mechanism. Theoretically, we establish convergence guarantees for SSLS in total variation (TV) distance, yielding concrete insights into the algorithm's error behavior with respect to key hyperparameters. Crucially, our derived error bounds demonstrate the asymptotic stability of SSLS, guaranteeing that local posterior sampling errors do not accumulate indefinitely over time. Extensive numerical experiments across challenging scenarios, including high-dimensional systems, strong nonlinearity, and sparse observations, highlight the robust performance of the proposed method. Furthermore, SSLS effectively quantifies the uncertainty associated with state estimates, rendering it particularly valuable for reliable error calibration.

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

0 major / 3 minor

Summary. The paper introduces score-based sequential Langevin sampling (SSLS) for nonlinear data assimilation in a recursive Bayesian filtering setting. It alternates dynamic-model prediction steps with score-based Langevin Monte Carlo updates that incorporate an annealing schedule to sample from non-log-concave posteriors. The central theoretical contribution is a set of total-variation error bounds that establish asymptotic stability, ensuring local sampling errors do not accumulate over time. Numerical experiments on high-dimensional, strongly nonlinear, and sparsely observed systems are presented to illustrate performance and uncertainty quantification.

Significance. If the TV bounds and their dependence on the annealing schedule are correctly established, the work supplies a theoretically grounded alternative to existing nonlinear assimilation schemes that also quantifies posterior uncertainty. The explicit stability result with respect to hyperparameters is a concrete strength.

minor comments (3)
  1. The abstract states that the annealing strategy overcomes challenges in non-log-concave sampling, but the precise schedule (temperature sequence, number of steps per temperature) is not summarized in the introduction; a short paragraph or table listing the schedule parameters used in the experiments would improve reproducibility.
  2. Notation for the score function and the Langevin dynamics is introduced without an explicit reference to the standard definition (e.g., the Ornstein-Uhlenbeck or overdamped Langevin SDE); adding one sentence with the SDE would clarify the update step for readers outside the score-based sampling community.
  3. Figure captions for the high-dimensional experiments should state the dimension, observation sparsity ratio, and the precise metric (RMSE, coverage, etc.) plotted; several captions currently omit these quantities.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of the manuscript. The recommendation for minor revision is appreciated, and we note that no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces SSLS as a new algorithmic decomposition into prediction and update steps with score-based Langevin Monte Carlo and annealing, then derives independent TV-distance convergence bounds showing asymptotic stability. No quoted equations or self-citations reduce any central claim (error bounds, stability guarantee) to fitted inputs, self-definitions, or prior author results by construction. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the method is described as building on existing score-based and Langevin techniques.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Rethinking Forward Processes for Score-Based Nonlinear Data Assimilation in High Dimensions

    stat.ML 2026-04 unverdicted novelty 6.0

    A measurement-aware forward process for score-based data assimilation yields an exact likelihood score for linear measurements by construction.

  2. Rethinking Forward Processes for Score-Based Nonlinear Data Assimilation in High Dimensions

    stat.ML 2026-04 unverdicted novelty 6.0

    MASF redesigns the forward diffusion process to align with measurements, yielding a theoretically grounded likelihood score and up to 28.2x speedup on O(10^5)-dimensional Kolmogorov flow under sparse and nonlinear obs...

Reference graph

Works this paper leans on

74 extracted references · 74 canonical work pages · cited by 1 Pith paper

  1. [1]

    Deep B ayesian inversion

    Jonas Adler and Ozan \"O ktem. Deep B ayesian inversion. In Tatiana A. Bubba, editor, Data-driven Models in Inverse Problems, volume 31 of Radon Series on Computational and Applied Mathematics, pages 359--412. Berlin, Boston: De Gruyter, 2025

    work page 2025

  2. [2]

    Anderson

    Jeffrey L. Anderson. Spatially and temporally varying adaptive covariance inflation for ensemble filters. Tellus A, 61 0 (1): 0 72--83, 2009

    work page 2009

  3. [3]

    Anderson and Stephen L

    Jeffrey L. Anderson and Stephen L. Anderson. A M onte C arlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Monthly Weather Review, 121: 0 2741--2758, 1999

    work page 1999

  4. [4]

    Uncertainty estimation for computed tomography with a linearised deep image prior

    Javier Antoran, Riccardo Barbano, Johannes Leuschner, Jos \'e Miguel Hern \'a ndez-Lobato, and Bangti Jin. Uncertainty estimation for computed tomography with a linearised deep image prior. Transactions on Machine Learning Research, 2023. ISSN 2835--8856

    work page 2023

  5. [5]

    Analysis and Geometry of Markov Diffusion Operators, volume 348 of Grundlehren der mathematischen Wissenschaften (GL)

    Dominique Bakr, Ivan Gentil, and Michel Ledoux. Analysis and Geometry of Markov Diffusion Operators, volume 348 of Grundlehren der mathematischen Wissenschaften (GL). Springer Cham, first edition, 2014

    work page 2014

  6. [6]

    A score-based filter for nonlinear data assimilation

    Feng Bao, Zezhong Zhang, and Guannan Zhang. A score-based filter for nonlinear data assimilation. Journal of Computational Physics, 514: 0 113207, 2024. ISSN 0021-9991

    work page 2024

  7. [7]

    Curse-of-dimensionality revisited: C ollapse of the particle filter in very large scale systems

    Thomas Bengtsson, Peter Bickel, and Bo Li. Curse-of-dimensionality revisited: C ollapse of the particle filter in very large scale systems. In Deborah Nolan and Terry Speed, editors, Institute of Mathematical Statistics Collections, Probability and Statistics: Essays in Honor of David A. Freedman, pages 316--334. Institute of Mathematical Statistics, 2008

    work page 2008

  8. [8]

    Muzaffer, and Andrew M

    Alexandros Beskos, Ajay Jasra, Ege A. Muzaffer, and Andrew M. Stuart. Sequential M onte C arlo methods for B ayesian elliptic inverse problems. Statistics and Computing, 25: 0 727--737, 2015

    work page 2015

  9. [9]

    Stochastic Filtering with Applications in Finance

    Ramaprasad Bhar. Stochastic Filtering with Applications in Finance. World Scientific, 2010

    work page 2010

  10. [10]

    Sharp failure rates for the bootstrap particle filter in high dimensions

    Peter Bickel, Bo Li, and Thomas Bengtsson. Sharp failure rates for the bootstrap particle filter in high dimensions. In Bertrand Clarke and Subhashis Ghosal, editors, Institute of Mathematical Statistics Collections, Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh, pages 318--329. Institute of Mathematical Statist...

    work page 2008

  11. [11]

    Evaluating raw ensembles with the continuous ranked probability score

    Jochen Br \"o cker. Evaluating raw ensembles with the continuous ranked probability score. Quarterly Journal of the Royal Meteorological Society, 138 0 (667): 0 1611--1617, 2012

    work page 2012

  12. [12]

    Normalizing constants of log-concave densities

    Nicolas Brosse, Alain Durmus, and \'E ric Moulines. Normalizing constants of log-concave densities. Electronic Journal of Statistics, 12 0 (1): 0 851 -- 889, 2018

    work page 2018

  13. [13]

    Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions

    Sitan Chen, Sinho Chewi, Jerry Li, Yuanzhi Li, Adil Salim, and Anru Zhang. Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions. In The Eleventh International Conference on Learning Representations, 2023

    work page 2023

  14. [14]

    Log-concave sampling, 2024

    Sinho Chewi. Log-concave sampling, 2024. URL https://chewisinho.github.io/main.pdf. unfinished draft

    work page 2024

  15. [15]

    Erdogdu, Mufan Li, Ruoqi Shen, and Matthew S

    Sinho Chewi, Murat A. Erdogdu, Mufan Li, Ruoqi Shen, and Matthew S. Zhang. Analysis of L angevin M onte C arlo from P oincar \'e to log- S obolev. Foundations of Computational Mathematics, 2024

    work page 2024

  16. [16]

    Diffusion posterior sampling for general noisy inverse problems

    Hyungjin Chung, Jeongsol Kim, Michael Thompson Mccann, Marc Louis Klasky, and Jong Chul Ye. Diffusion posterior sampling for general noisy inverse problems. In The Eleventh International Conference on Learning Representations, 2023

    work page 2023

  17. [17]

    Bayesian inverse problems for functions and applications to fluid mechanics

    S L Cotter, M Dashti, J C Robinson, and A M Stuart. Bayesian inverse problems for functions and applications to fluid mechanics. Inverse Problems, 25 0 (11): 0 115008, 2009

    work page 2009

  18. [18]

    Sequential M onte C arlo samplers

    Pierre Del Moral, Arnaud Doucet, and Ajay Jasra. Sequential M onte C arlo samplers. Journal of the Royal Statistical Society Series B: Statistical Methodology, 68 0 (3): 0 411--436, 05 2006

    work page 2006

  19. [19]

    Sequential Monte Carlo Methods in Practice

    Arnaud Doucet, Nando Freitas, and Neil Gordon, editors. Sequential Monte Carlo Methods in Practice. Information Science and Statistics (ISS). Springer New York, NY, first edition, 2001

    work page 2001

  20. [20]

    John C. Duchi. Information theory and statistics, 2024. URL http://web.stanford.edu/class/stats311/lecture-notes.pdf. unfinished draft

    work page 2024

  21. [21]

    Elliott and Tak Kuen Siu

    Robert J. Elliott and Tak Kuen Siu. Option pricing and filtering with hidden M arkov-modulated pure-jump processes. Applied Mathematical Finance, 20 0 (1): 0 1--25, 2013

    work page 2013

  22. [22]

    Vossepoel, and Peter Jan van Leeuwen

    Geir Evensen, Femke C. Vossepoel, and Peter Jan van Leeuwen. Data Assimilation Fundamentals: A Unified Formulation of the State and Parameter Estimation Problem. Springer Textbooks in Earth Sciences, Geography and Environment (STEGE). Springer Cham, first edition, 2022

    work page 2022

  23. [23]

    Pricing and hedging of credit derivatives via the innovations approach to nonlinear filtering

    R \"u diger Frey and Thorsten Schmidt. Pricing and hedging of credit derivatives via the innovations approach to nonlinear filtering. Finance and Stochastics, 16: 0 105--133, 2012

    work page 2012

  24. [24]

    Estimating normalizing constants for log-concave distributions: algorithms and lower bounds

    Rong Ge, Holden Lee, and Jianfeng Lu. Estimating normalizing constants for log-concave distributions: algorithms and lower bounds. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, pages 579--586. Association for Computing Machinery, 2020

    work page 2020

  25. [25]

    Tilmann Gneiting, Fadoua Balabdaoui, and Adrian E. Raftery. Probabilistic forecasts, calibration and sharpness. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 69 0 (2): 0 243--268, 2007

    work page 2007

  26. [26]

    Gordon, D.J

    N.J. Gordon, D.J. Salmond, and A.F.M. Smith. Novel approach to nonlinear/non- G aussian B ayesian state estimation. IEE Proceedings on Radar and Signal Processing, 140: 0 107--113, 1993

    work page 1993

  27. [27]

    Gradient guidance for diffusion models: A n optimization perspective, 2024

    Yingqing Guo, Hui Yuan, Yukang Yang, Minshuo Chen, and Mengdi Wang. Gradient guidance for diffusion models: A n optimization perspective, 2024. arXiv:2404.14743

    work page arXiv 2024

  28. [28]

    Denoising diffusion probabilistic models

    Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 6840--6851. Curran Associates, Inc., 2020

    work page 2020

  29. [29]

    Data assimilation using an ensemble kalman filter technique

    Peter L Houtekamer and Herschel L Mitchell. Data assimilation using an ensemble kalman filter technique. Monthly weather review, 126 0 (3): 0 796--811, 1998

    work page 1998

  30. [30]

    Estimation of non-normalized statistical models by score matching

    Aapo Hyv \"a rinen. Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research, 6 0 (24): 0 695--709, 2005

    work page 2005

  31. [31]

    Robust compressed sensing MRI with deep generative priors

    Ajil Jalal, Marius Arvinte, Giannis Daras, Eric Price, Alexandros G Dimakis, and Jon Tamir. Robust compressed sensing MRI with deep generative priors. In M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, editors, Advances in Neural Information Processing Systems, volume 34, pages 14938--14954. Curran Associates, Inc., 2021

    work page 2021

  32. [32]

    Deep nonparametric regression on approximate manifolds: Nonasymptotic error bounds with polynomial prefactors

    Yuling Jiao, Guohao Shen, Yuanyuan Lin, and Jian Huang. Deep nonparametric regression on approximate manifolds: Nonasymptotic error bounds with polynomial prefactors. The Annals of Statistics, 51 0 (2): 0 691 -- 716, 2023

    work page 2023

  33. [33]

    Sequential M onte C arlo methods for high-dimensional inverse problems: A case study for the N avier- S tokes equations

    Nikolas Kantas, Alexandros Beskos, and Ajay Jasra. Sequential M onte C arlo methods for high-dimensional inverse problems: A case study for the N avier- S tokes equations. SIAM/ASA Journal on Uncertainty Quantification, 2 0 (1): 0 464--489, 2014

    work page 2014

  34. [34]

    Numerical Weather Prediction and Data Assimilation

    Petros Katsafados, Elias Mavromatidis, and Christos Spyrou. Numerical Weather Prediction and Data Assimilation. John Wiley & Sons, Ltd, 2020

    work page 2020

  35. [35]

    Monte C arlo filter and smoother for non- G aussian nonlinear state space models

    Genshiro Kitagawa. Monte C arlo filter and smoother for non- G aussian nonlinear state space models. Journal of Computational and Graphical Statistics, 5 0 (1): 0 1--25, 1996

    work page 1996

  36. [36]

    Smith, Ayya Alieva, Qing Wang, Michael P

    Dmitrii Kochkov, Jamie A. Smith, Ayya Alieva, Qing Wang, Michael P. Brenner, and Stephan Hoyer. Machine learning-accelerated computational fluid dynamics. Proceedings of the National Academy of Sciences, 118 0 (21): 0 e2101784118, 2021

    work page 2021

  37. [37]

    On the rate of convergence of fully connected deep neural network regression estimates

    Michael Kohler and Sophie Langer. On the rate of convergence of fully connected deep neural network regression estimates. The Annals of Statistics, 49 0 (4): 0 2231--2249, 2021

    work page 2021

  38. [38]

    Data Assimilation: A Mathematical Introduction, volume 62 of Texts in Applied Mathematics (TAM)

    Kody Law, Andrew Stuart, and Konstantinos Zygalakis. Data Assimilation: A Mathematical Introduction, volume 62 of Texts in Applied Mathematics (TAM). Springer Cham, first edition, 2015

    work page 2015

  39. [39]

    Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects

    Fran c ois-Xavier Le Dimet and Olivier Talagrand. Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus A, 38A 0 (2): 0 97--110, 1986

    work page 1986

  40. [40]

    Convergence for score-based generative modeling with polynomial complexity

    Holden Lee, Jianfeng Lu, and Yixin Tan. Convergence for score-based generative modeling with polynomial complexity. In S. Koyejo, S. Mohamed, A. Agarwal, D. Belgrave, K. Cho, and A. Oh, editors, Advances in Neural Information Processing Systems, volume 35, pages 22870--22882. Curran Associates, Inc., 2022

    work page 2022

  41. [41]

    State-observation augmented diffusion model for nonlinear assimilation, 2024

    Zhuoyuan Li, Bin Dong, and Pingwen Zhang. State-observation augmented diffusion model for nonlinear assimilation, 2024. arXiv:2407.21314

    work page arXiv 2024

  42. [42]

    Majda and John Harlim

    Andrew J. Majda and John Harlim. Filtering Complex Turbulent Systems. Cambridge University Press, 2012

    work page 2012

  43. [43]

    Jan Mandel, Loren Cobb, and Jonathan D. Beezley. On the convergence of the ensemble kalman filter. Applications of Mathematics, 56: 0 533--541, 2012

    work page 2012

  44. [44]

    Patel and Assad A

    Dhruv V. Patel and Assad A. Oberai. GAN -based priors for quantifying uncertainty in supervised learning. SIAM/ASA Journal on Uncertainty Quantification, 9 0 (3): 0 1314--1343, 2021

    work page 2021

  45. [45]

    Pavliotis

    Grigorios A. Pavliotis. Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations, volume 60 of Texts in Applied Mathematics (TAM). Springer New York, NY, 2014

    work page 2014

  46. [46]

    Filtering via simulation: Auxiliary particle filters

    Michael K Pitt and Neil Shephard. Filtering via simulation: Auxiliary particle filters. Journal of the American statistical association, 94 0 (446): 0 590--599, 1999

    work page 1999

  47. [47]

    Posterior sampling via L angevin dynamics based on generative priors, 2024

    Vishal Purohit, Matthew Repasky, Jianfeng Lu, Qiang Qiu, Yao Xie, and Xiuyuan Cheng. Posterior sampling via L angevin dynamics based on generative priors, 2024. arXiv:2410.02078

    work page arXiv 2024

  48. [48]

    Ragheb Raad, Dhruv Patel, Chiao-Chih Hsu, Vijay Kothapalli, Deep Ray, Bino Varghese, Darryl Hwang, Inderbir Gill, Vinay Duddalwar, and Assad A. Oberai. Probabilistic medical image imputation via deep adversarial learning. Engineering with Computers, 38: 0 3975--3986, 2022

    work page 2022

  49. [49]

    Ragheb Raad, Deep Ray, Bino Varghese, Darryl Hwang, Inderbir Gill, Vinay Duddalwar, and Assad A. Oberai. Conditional generative learning for medical image imputation. Scientific Reports, 14 0 (171), 2024

    work page 2024

  50. [50]

    A Course on Large Deviations with an Introduction to Gibbs Measures, volume 162 of Graduate Studies in Mathematics

    Firas Rassoul-Agha and Timo Sepp \"a l \"a inen. A Course on Large Deviations with an Introduction to Gibbs Measures, volume 162 of Graduate Studies in Mathematics. American Mathematical Society (AMS), 2015

    work page 2015

  51. [51]

    Data assimilation: T he S chr \"o dinger perspective

    Sebastian Reich. Data assimilation: T he S chr \"o dinger perspective. Acta Numerica, 28: 0 635--711, 2019

    work page 2019

  52. [52]

    Probabilistic Forecasting and Bayesian Data Assimilation

    Sebastian Reich and Colin Cotter. Probabilistic Forecasting and Bayesian Data Assimilation. Cambridge University Press, 2015

    work page 2015

  53. [53]

    Score-based data assimilation

    Fran c ois Rozet and Gilles Louppe. Score-based data assimilation. In A. Oh, T. Naumann, A. Globerson, K. Saenko, M. Hardt, and S. Levine, editors, Advances in Neural Information Processing Systems, volume 36, pages 40521--40541. Curran Associates, Inc., 2023 a

    work page 2023

  54. [54]

    Score-based data assimilation for a two-layer quasi-geostrophic model, 2023 b

    Fran c ois Rozet and Gilles Louppe. Score-based data assimilation for a two-layer quasi-geostrophic model, 2023 b . arXiv:2310.01853

    work page arXiv 2023

  55. [55]

    Sampling errors in ensemble K alman filtering

    William Sacher and Peter Bartello. Sampling errors in ensemble K alman filtering. P art I : T heory. Monthly Weather Review, 136: 0 3035--3049, 2008

    work page 2008

  56. [56]

    Bayesian filtering and smoothing

    Simo S \"a rkk \"a and Lennart Svensson. Bayesian filtering and smoothing. Institute of Mathematical Statistics Textbooks. Cambridge University Press, second edition, 2023

    work page 2023

  57. [57]

    Nonparametric regression using deep neural networks with relu activation function

    Johannes Schmidt-Hieber. Nonparametric regression using deep neural networks with relu activation function. The Annals of Statistics, 48 0 (4): 0 1875--1897, 2020

    work page 2020

  58. [58]

    Latent-EnSF : A latent ensemble score filter for high-dimensional data assimilation with sparse observation data, 2024

    Phillip Si and Peng Chen. Latent-EnSF : A latent ensemble score filter for high-dimensional data assimilation with sparse observation data, 2024. arXiv:2409.00127

    work page arXiv 2024

  59. [59]

    Obstacles to high-dimensional particle filtering

    Chris Snyder, Thomas Bengtsson, Peter Bickel, and Jeff Anderson. Obstacles to high-dimensional particle filtering. Monthly weather review, 136: 0 4629--4640, 2008

    work page 2008

  60. [60]

    Pseudoinverse-guided diffusion models for inverse problems

    Jiaming Song, Arash Vahdat, Morteza Mardani, and Jan Kautz. Pseudoinverse-guided diffusion models for inverse problems. In International Conference on Learning Representations, 2023 a

    work page 2023

  61. [61]

    Loss-guided diffusion models for plug-and-play controllable generation

    Jiaming Song, Qinsheng Zhang, Hongxu Yin, Morteza Mardani, Ming-Yu Liu, Jan Kautz, Yongxin Chen, and Arash Vahdat. Loss-guided diffusion models for plug-and-play controllable generation. In Andreas Krause, Emma Brunskill, Kyunghyun Cho, Barbara Engelhardt, Sivan Sabato, and Jonathan Scarlett, editors, Proceedings of the 40th International Conference on Ma...

    work page 2023

  62. [62]

    Generative modeling by estimating gradients of the data distribution

    Yang Song and Stefano Ermon. Generative modeling by estimating gradients of the data distribution. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d Alch\' e -Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019

    work page 2019

  63. [63]

    Sliced score matching: A scalable approach to density and score estimation

    Yang Song, Sahaj Garg, Jiaxin Shi, and Stefano Ermon. Sliced score matching: A scalable approach to density and score estimation. In Ryan P. Adams and Vibhav Gogate, editors, Proceedings of The 35th Uncertainty in Artificial Intelligence Conference, volume 115 of Proceedings of Machine Learning Research, pages 574--584. PMLR, 22--25 Jul 2020

    work page 2020

  64. [64]

    Score-based generative modeling through stochastic differential equations

    Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021

    work page 2021

  65. [65]

    Coupling techniques for nonlinear ensemble filtering

    Alessio Spantini, Ricardo Baptista, and Youssef Marzouk. Coupling techniques for nonlinear ensemble filtering. SIAM Review, 64 0 (4): 0 921--953, 2022

    work page 2022

  66. [66]

    Sullivan

    T.J. Sullivan. Introduction to Uncertainty Quantification, volume 63 of Texts in Applied Mathematics (TAM). Springer Cham, first edition, 2015

    work page 2015

  67. [67]

    Adaptivity of diffusion models to manifold structures

    Rong Tang and Yun Yang. Adaptivity of diffusion models to manifold structures. In Sanjoy Dasgupta, Stephan Mandt, and Yingzhen Li, editors, Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, volume 238 of Proceedings of Machine Learning Research, pages 1648--1656. PMLR, 02--04 May 2024

    work page 2024

  68. [68]

    Byeng Youn, Michael D

    Adam Thelen, Xiaoge Zhang, Olga Fink, Yan Lu, Sayan Ghosh, D. Byeng Youn, Michael D. Todd, Sankaran Mahadevan, Chao Hu, and Zhen Hu. A comprehensive review of digital twin -- part 1: modeling and twinning enabling technologies. Structural and Multidisciplinary Optimization, 65 0 (354), 2022

    work page 2022

  69. [69]

    Byeng Youn, Michael D

    Adam Thelen, Xiaoge Zhang, Olga Fink, Yan Lu, Sayan Ghosh, D. Byeng Youn, Michael D. Todd, Sankaran Mahadevan, Chao Hu, and Zhen Hu. A comprehensive review of digital twin -- part 2: roles of uncertainty quantification and optimization, a battery digital twin, and perspectives. Structural and Multidisciplinary Optimization, 66 0 (1), 2023

    work page 2023

  70. [70]

    Tsybakov

    Alexandre B. Tsybakov. Introduction to Nonparametric Estimation. Springer Series in Statistics (SSS). Springer New York, NY, first edition, 2009

    work page 2009

  71. [71]

    Rapid convergence of the unadjusted L angevin algorithm: Isoperimetry suffices

    Santosh Vempala and Andre Wibisono. Rapid convergence of the unadjusted L angevin algorithm: Isoperimetry suffices. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d Alch\' e -Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019

    work page 2019

  72. [72]

    A connection between score matching and denoising autoencoders

    Pascal Vincent. A connection between score matching and denoising autoencoders. Neural Computation, 23 0 (7): 0 1661--1674, 2011

    work page 2011

  73. [73]

    Wainwright

    Martin J. Wainwright. High-Dimensional Statistics: A Non-Asymptotic Viewpoint. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, 2019

    work page 2019

  74. [74]

    Annealing flow generative model towards sampling high-dimensional and multi-modal distributions, 2024

    Dongze Wu and Yao Xie. Annealing flow generative model towards sampling high-dimensional and multi-modal distributions, 2024. arXiv:2409.20547

    work page arXiv 2024