pith. sign in

arxiv: 2508.10630 · v2 · submitted 2025-08-14 · 🧮 math.NA · cs.NA· stat.CO· stat.ML

Nonlinear filtering based on density approximation and deep BSDE prediction

Kasper B{\aa}gmark , Adam Andersson , Stig Larsson This is my paper

Pith reviewed 2026-05-18 23:13 UTC · model grok-4.3

classification 🧮 math.NA cs.NAstat.COstat.ML
keywords nonlinear filteringBayesian filterdeep BSDEFeynman-Kac representationdensity approximationerror boundsHörmander conditionnumerical convergence
0
0 comments X

The pith

A new approximate Bayesian filter uses nonlinear Feynman-Kac representation and deep BSDE to approximate the unnormalized filtering density with a hybrid error bound.

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

The paper introduces an approximate Bayesian filter for nonlinear stochastic systems that relies on a nonlinear Feynman-Kac representation of the filtering problem. It approximates the unnormalized filtering density offline using the deep BSDE method and neural networks so that the filter can run online with fresh observations. A hybrid a priori-a posteriori error bound is established when the underlying diffusion satisfies a parabolic Hörmander condition, and numerical examples confirm that the observed convergence rate matches the theoretical prediction.

Core claim

The paper establishes a novel approximate Bayesian filter that employs a nonlinear Feynman-Kac representation of the filtering problem together with the deep BSDE method and neural networks to approximate the unnormalized filtering density. The resulting scheme is trained offline and then applied online to new observations. Under the parabolic Hörmander condition a hybrid a priori-a posteriori error bound is proved, and the predicted convergence rate is verified numerically on two test problems.

What carries the argument

Nonlinear Feynman-Kac representation combined with deep BSDE approximation of the unnormalized filtering density; this representation converts the filtering density evolution into a backward stochastic differential equation that neural networks can solve offline.

Load-bearing premise

The underlying stochastic system must satisfy the parabolic Hörmander condition so that the hybrid error bound for the density approximation holds.

What would settle it

Run the method on a diffusion whose generator fails the parabolic Hörmander condition and check whether the observed approximation error still decays at the rate predicted by the hybrid bound.

Figures

Figures reproduced from arXiv: 2508.10630 by Adam Andersson, Kasper B{\aa}gmark, Stig Larsson.

Figure 1
Figure 1. Figure 1: The figure illustrates the flowchart of the chain of prediction and up￾date steps through out the method. Each Deep BSDE box depicts a BSDE and solving it through the optimization problem (15). This system of equations is the same as (10) except for the terminal conditions of the Ye k processes and we note that X = Xe. By the a priori estimate [17, Proposition 2.1], there exists a constant C1 > 0 so that s… view at source ↗
Figure 2
Figure 2. Figure 2: The figure illustrates the numerical scheme for a fixed k = 1, . . . , K −1 in the optimization problem (27). The color of the arrows between the boxes indicate if the map consist of an Euler–Maruyama step, a parameterized neural network step or simply an input to a function. 4.2. Numerical experiments. In this section we empirically examine the numerical convergence of our approximative filter applied to … view at source ↗
Figure 3
Figure 3. Figure 3: Ornstein–Uhlenbeck process. The figure illustrates the error trajecto￾ries for seven different discretizations. To the left we see the error ek and to the right we see the residual Ek, k = 1, . . . , K. To investigate the convergence rate, we fix k = K and plot both the final time error eK and the accumulated a posteriori term E := PK k=1 Ek in [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Ornstein–Uhlenbeck process. The figure presents the error and the accumulated residual at the final time for seven different discretizations together with a reference line of order 1 2 . To the left we have eK and to the right the accumulated residual E = PK k=1 Ek. Bistable process. The second example concerns an SDE with nonlinear drift given by µ(x) = 2 5 (5x − x 3 ). The drift corresponds to the gradie… view at source ↗
Figure 5
Figure 5. Figure 5: Bistable process. The figure illustrates the error trajectories for seven different discretizations. To the left we see the error ek and to the right we see the residual Ek, k = 1, . . . , K. finest discretization. In contrast, the a posteriori term E follows the anticipated order 1 2 decay up to N = 16, after which the convergence stagnates. This behavior aligns with the theoretical structure of the error… view at source ↗
Figure 6
Figure 6. Figure 6: Bistable process. The figure presents the error and the accumulated residual at the final time for seven different discretizations together with a refer￾ence line of order 1 2 . To the left we have eK and to the right the accumulated residual E = PK k=1 Ek. 5. Conclusion and outlook In this paper, we introduce an approximative method for the filtering problem in a continuous￾discrete setting, prove an erro… view at source ↗
read the original abstract

A novel approximate Bayesian filter based on backward stochastic differential equations is introduced. It uses a nonlinear Feynman--Kac representation of the filtering problem and the approximation of an unnormalized filtering density using the well-known deep BSDE method and neural networks. The method is trained offline, which means that it can be applied online with new observations. A hybrid a priori-a posteriori error bound is proved under a parabolic H\"ormander condition. The theoretical convergence rate is confirmed in two numerical examples.

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 introduces a novel approximate Bayesian filter that uses a nonlinear Feynman-Kac representation of the filtering problem and approximates the unnormalized filtering density via the deep BSDE method with neural networks. The approach is trained offline for online application with new observations. A hybrid a priori-a posteriori error bound is proved under a parabolic Hörmander condition, and the theoretical convergence rate is confirmed numerically in two examples.

Significance. If the hybrid error bound is valid and the numerical confirmation is robust, the work provides a theoretically supported method for nonlinear filtering that combines BSDE approximations with density estimation. This could be useful for high-dimensional problems. The offline training aspect is a practical advantage, and the hybrid bound represents a constructive attempt to blend a priori analysis with a posteriori control. The numerical examples offer some empirical support for the claimed rate.

major comments (2)
  1. [Abstract and theoretical analysis section] Abstract and theoretical analysis section: The hybrid a priori-a posteriori error bound is proved under the parabolic Hörmander condition to control the density approximation error inside the nonlinear Feynman-Kac representation. However, the manuscript provides no verification that the driving SDEs in the two numerical examples satisfy the Lie-algebra rank condition on the relevant time interval. This assumption is load-bearing for the claimed convergence rate, as its failure would collapse the a-priori component of the bound.
  2. [Section on deep BSDE approximation and error analysis] Section on deep BSDE approximation and error analysis: The paper does not demonstrate that the neural-network error from the deep BSDE training step inherits the hypoelliptic regularity supplied by the parabolic Hörmander condition. Without this link, it is unclear whether the practical algorithm inherits the theoretical rate, even if the underlying SDE satisfies the assumption.
minor comments (2)
  1. [Method section] The definition of the unnormalized filtering density and its relation to the nonlinear Feynman-Kac representation could be stated more explicitly at the beginning of the method section for improved readability.
  2. [Numerical examples] Numerical example figures would benefit from additional details in the captions regarding the specific parameters, time horizons, and network architectures used to confirm the convergence rate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We address each point below and indicate the revisions that will be incorporated in the next version.

read point-by-point responses

  1. Referee: [Abstract and theoretical analysis section] Abstract and theoretical analysis section: The hybrid a priori-a posteriori error bound is proved under the parabolic Hörmander condition to control the density approximation error inside the nonlinear Feynman-Kac representation. However, the manuscript provides no verification that the driving SDEs in the two numerical examples satisfy the Lie-algebra rank condition on the relevant time interval. This assumption is load-bearing for the claimed convergence rate, as its failure would collapse the a-priori component of the bound.

    Authors: We agree that an explicit verification of the Lie-algebra rank condition for the driving SDEs in the numerical examples strengthens the link between theory and experiments. Both examples were selected from the standard nonlinear filtering literature precisely because they satisfy the parabolic Hörmander condition on the intervals considered; the first is uniformly elliptic and the second has a diffusion coefficient whose Lie brackets span the tangent space. In the revised manuscript we will add a short paragraph (or subsection) that recalls the relevant vector fields and confirms the rank condition holds, thereby making the applicability of the hybrid bound fully transparent. revision: yes

  2. Referee: [Section on deep BSDE approximation and error analysis] Section on deep BSDE approximation and error analysis: The paper does not demonstrate that the neural-network error from the deep BSDE training step inherits the hypoelliptic regularity supplied by the parabolic Hörmander condition. Without this link, it is unclear whether the practical algorithm inherits the theoretical rate, even if the underlying SDE satisfies the assumption.

    Authors: The hybrid bound separates the error into an a-priori term that exploits the hypoelliptic regularity to control the density approximation inside the nonlinear Feynman-Kac representation and an a-posteriori term that controls the deep-BSDE neural-network training error via standard empirical-risk bounds. The regularity guarantees that the associated PDE solution is sufficiently smooth for the BSDE to be well-defined and for the neural-network approximation theory to apply with the expected rate. To make this inheritance explicit we will insert a clarifying remark in the error-analysis section that recalls how hypoelliptic regularity propagates to the BSDE solution and thereby justifies the use of the same convergence rate for the neural-network component. This addition will remove any ambiguity about whether the practical algorithm inherits the theoretical rate. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external Hörmander condition and standard deep BSDE

full rationale

The paper's central derivation introduces a nonlinear Feynman-Kac representation of the filtering problem, approximates the unnormalized density via the established deep BSDE method with neural networks, and proves a hybrid a priori-a posteriori error bound under the parabolic Hörmander condition. This condition is a standard external assumption from stochastic analysis (hypoelliptic regularity) rather than a self-derived or self-cited result. The deep BSDE training is offline and uses well-known techniques without any fitted parameters being renamed as predictions or any self-definitional loops in the equations. Numerical examples confirm rates but do not constitute the theoretical claim. No load-bearing step reduces by construction to the paper's own inputs; the argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method depends on standard stochastic analysis assumptions and neural network training; the Hörmander condition is the key domain assumption enabling the error analysis.

free parameters (1)
  • Neural network weights and biases
    Parameters of the networks approximating the BSDE solution are fitted during offline training to data generated from the model.
axioms (1)
  • domain assumption Parabolic Hörmander condition on the diffusion coefficients
    Invoked to prove the hybrid a priori-a posteriori error bound for the density approximation.

pith-pipeline@v0.9.0 · 5610 in / 1371 out tokens · 31109 ms · 2026-05-18T23:13:08.054814+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

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

  1. High-dimensional Bayesian filtering through deep density approximation

    math.NA 2025-11 unverdicted novelty 5.0

    The logarithmic deep backward SDE filter succeeds in a 100-dimensional Lorenz-96 model where particle and ensemble Kalman filters fail, while cutting inference time by two to five orders of magnitude.

Reference graph

Works this paper leans on

50 extracted references · 50 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Andersson, A

    K. Andersson, A. Andersson, and C. W. Oosterlee. The deep multi-FBSDE method: a robust deep learning method for coupled FBSDEs. arXiv:2503.13193, 2025

    work page arXiv 2025

  2. [2]

    B˚ agmark, A

    K. B˚ agmark, A. Andersson, and S. Larsson. An energy-based deep splitting method for the nonlinear filtering problem. Partial Differ. Equ. Appl. , 4, 2023

    work page 2023

  3. [3]

    A convergent scheme for the Bayesian filtering problem based on the Fokker--Planck equation and deep splitting

    K. B˚ agmark, A. Andersson, S. Larsson, and F. Rydin. A convergent scheme for the Bayesian filtering problem based on the Fokker–Planck equation and deep splitting. arXiv:2409.14585, 2024

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  4. [4]

    Bar-Shalom, X

    Y. Bar-Shalom, X. R. Li, and T. Kirubarajan. Estimation with Applications to Tracking and Navigation . John Wiley & Sons, 2001

    work page 2001

  5. [5]

    C. Beck, S. Becker, P. Cheridito, A. Jentzen, and A. Neufeld. Deep learning based numerical approxima- tion algorithms for stochastic partial differential equations and high-dimensional nonlinear filtering problems. arXiv:2012.01194, 2020. NONLINEAR FILTERING BASED ON DENSITY APPROXIMATION AND DEEP BSDE PREDICTION 17

    work page arXiv 2012

  6. [6]

    C. Beck, S. Becker, P. Cheridito, A. Jentzen, and A. Neufeld. Deep splitting method for parabolic PDEs. SIAM J. Sci. Comput. , 43:A3135–A3154, 2021

    work page 2021

  7. [7]

    S. S. Blackman and R. Popoli. Design and Analysis of Modern Tracking Systems . Artech House Publishers, 1999

    work page 1999

  8. [8]

    Challa and Y

    S. Challa and Y. Bar-Shalom. Nonlinear filter design using Fokker-Planck-Kolmogorov probability density evolutions. IEEE Trans. Aerosp. Electron. Syst., 36:309–315, 2000

    work page 2000

  9. [9]

    Chopin, A

    N. Chopin, A. Fulop, J. Heng, and A. H. Thiery. Computational Doob h-transforms for online filtering of discretely observed diffusions. In Int. Conf. Mach. Learn. , pages 5904–5923. PMLR, 2023

    work page 2023

  10. [10]

    Corenflos and A

    A. Corenflos and A. Finke. Particle-MALA and Particle-mGRAD: Gradient-based MCMC methods for high- dimensional state-space models. arXiv:2401.14868, 2024

    work page arXiv 2024

  11. [11]

    Corenflos, Z

    A. Corenflos, Z. Zhao, T. B. Sch¨ on, S. S¨ arkk¨ a, and J. Sj¨ olund. Conditioning diffusion models by explicit forward-backward bridging. In Int. Conf. Artif. Intell. Stat. , pages 3709–3717. PMLR, 2025

    work page 2025

  12. [12]

    Crisan, A

    D. Crisan, A. Lobbe, and S. Ortiz-Latorre. An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations. Stoch. Partial Differ. Equ.: Anal. Comput., 10:1050–1081, 2022

    work page 2022

  13. [13]

    N. Cui, L. Hong, and J. R. Layne. A comparison of nonlinear filtering approaches with an application to ground target tracking. Signal Processing, 85:1469–1492, 2005

    work page 2005

  14. [14]

    Demissie, M

    B. Demissie, M. A. Khan, and F. Govaers. Nonlinear filter design using Fokker-Planck propagator in Kronecker tensor format. In 2016 19th International Conference on Information Fusion (FUSION) , pages 1–8. IEEE, 2016

    work page 2016

  15. [15]

    W. E, J. Han, and A. Jentzen. Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Commun. Math. Stat , 5:349–380, Nov. 2017

    work page 2017

  16. [16]

    W. E and B. Yu. The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems. Commun. Math. Stat , 1:1–12, 2018

    work page 2018

  17. [17]

    El Karoui, S

    N. El Karoui, S. Peng, and M. C. Quenez. Backward stochastic differential equations in finance. Math. Finance, 7(1):1–71, 1997

    work page 1997

  18. [18]

    G. Evensen. Data Assimilation: The Ensemble Kalman Filter . Springer, 2009

    work page 2009

  19. [19]

    Finke and A

    A. Finke and A. H. Thiery. Conditional sequential Monte Carlo in high dimensions. Ann. Statist., 51:437–463, 2023

    work page 2023

  20. [20]

    M. B. Giles. Multilevel Monte Carlo methods. Acta Numerica, 24:259–328, 2015

    work page 2015

  21. [21]

    I. R. Goodman, R. P. S. Mahler, and H. T. Nguyen. Mathematics of Data Fusion , volume 37 of Theory and Decision Library. Series B: Mathematical and Statistical Methods . Kluwer Academic Publishers Group, Dordrecht, 1997

    work page 1997

  22. [22]

    N. J. Gordon, D. J. Salmond, and A. F. M. Smith. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings F (Radar and Signal Processing) , 140(2):107–113, 1993

    work page 1993

  23. [23]

    F. K. Gustafsson, M. Danelljan, G. Bhat, and T. B. Sch¨ on. Energy-based models for deep probabilistic regres- sion. In European Conference on Computer Vision , pages 325–343. Springer, 2020

    work page 2020

  24. [24]

    J. Han, W. Hu, J. Long, and Y. Zhao. Deep Picard iteration for high-dimensional nonlinear PDEs. arXiv:2409.08526, 2024

    work page arXiv 2024

  25. [25]

    Han and J

    J. Han and J. Long. Convergence of the deep BSDE method for coupled FBSDEs. Probab. Uncertain. Quant. Risk, 5:Paper No. 5, 33, 2020

    work page 2020

  26. [26]

    Heinrich

    S. Heinrich. Monte Carlo complexity of global solution of integral equations. J. Complexity , 14(2):151–175, 1998

    work page 1998

  27. [27]

    J. Ho, A. Jain, and P. Abbeel. Denoising diffusion probabilistic models.Adv. Neural Inf. Process. Syst., 33:6840– 6851, 2020

    work page 2020

  28. [28]

    Isard and A

    M. Isard and A. Blake. Condensation—conditional density propagation for visual tracking. Int. J. Comput. Vis., 29(1):5–28, 1998

    work page 1998

  29. [29]

    Jasra, D

    A. Jasra, D. A. Stephens, and C. C. Holmes. Population-Based Reversible Jump Markov Chain Monte Carlo. Biometrika, 92(4):803–820, 2005

    work page 2005

  30. [30]

    M. S. Johannes and N. G. Polson. MCMC Methods for Continuous-Time Financial Econometrics. In Handbook of Financial Econometrics , pages 1–72. Elsevier, 2009

    work page 2009

  31. [31]

    R. E. Kalman and R. S. Bucy. New results in linear filtering and prediction theory. J. Basic Eng. , 83:95–108, 1961

    work page 1961

  32. [32]

    Kapllani and L

    L. Kapllani and L. Teng. A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations. IMA J. Numer. Anal. , 2025

    work page 2025

  33. [33]

    K. P. K¨ ording and D. M. Wolpert. Bayesian integration in sensorimotor learning. Nature, 427(6971):244–247, 2004

    work page 2004

  34. [34]

    Krishnapriyan, A

    A. Krishnapriyan, A. Gholami, S. Zhe, R. Kirby, and M. W. Mahoney. Characterizing possible failure modes in physics-informed neural networks. Adv. Neural Inf. Process. Syst. , 34:26548–26560, 2021

    work page 2021

  35. [35]

    L. Lu, P. Jin, G. Pang, Z. Zhang, and G. E. Karniadakis. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nat. Mach. Intell. , 3:218–229, 2021

    work page 2021

  36. [36]

    K. Luo, J. Zhao, Y. Wang, J. Li, J. Wen, J. Liang, H. Soekmadji, and S. Liao. Physics-informed neural networks for PDE problems: a comprehensive review. Artif. Intell. Rev. , 58(10):1–43, 2025

    work page 2025

  37. [37]

    P. S. Maybeck. Stochastic Models, Estimation, and Control, Volume 1 . Academic Press, 1979

    work page 1979

  38. [38]

    C. A. Naesseth, F. Lindsten, and T. B. Sch¨ on. High-dimensional filtering using nested sequential Monte Carlo. IEEE Trans. Signal Process., 67:4177–4188, 2019. 18 K. B ˚AGMARK, A. ANDERSSON, AND S. LARSSON

    work page 2019

  39. [39]

    Øksendal

    B. Øksendal. Stochastic Differential Equations: An Introduction with Applications. Springer Science & Business Media, 2003

    work page 2003

  40. [40]

    Raissi, P

    M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. , 378:686–707, 2019

    work page 2019

  41. [41]

    Rombach, A

    R. Rombach, A. Blattmann, D. Lorenz, P. Esser, and B. Ommer. High-resolution image synthesis with latent diffusion models. In Proc. IEEE/CVF Conf. Comput. Vis. Pattern Recognit. , pages 10684–10695, 2022

    work page 2022

  42. [42]

    Ronneberger, P

    O. Ronneberger, P. Fischer, and T. Brox. U-Net: Convolutional Networks for Biomedical Image Segmentation. In Med. Image Comput. Comput. Assist. Interv. , pages 234–241. Springer, 2015

    work page 2015

  43. [43]

    Silverman

    B. Silverman. Density Estimation for Statistics and Data Analysis . Chapman & Hall/CRC, 1986

    work page 1986

  44. [44]

    C. Snyder. Particle filters, the “optimal” proposal and high-dimensional systems. In Proceedings of the ECMWF Seminar on Data Assimilation for atmosphere and ocean , pages 1–10, 2011

    work page 2011

  45. [45]

    Snyder, T

    C. Snyder, T. Bengtsson, and M. Morzfeld. Performance bounds for particle filters using the optimal proposal. Mon. Weather Rev., 143:4750–4761, 2015

    work page 2015

  46. [46]

    Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole. Score-based generative modeling through stochastic differential equations. In Int. Conf. Learn. Represent., 2021

    work page 2021

  47. [47]

    Thrun, W

    S. Thrun, W. Burgard, and D. Fox. Probabilistic Robotics. MIT Press, 2005

    work page 2005

  48. [48]

    van der Meulen and M

    F. van der Meulen and M. Schauer. Automatic backward filtering forward guiding for Markov processes and graphical models. arXiv:2010.03509, 2020

    work page arXiv 2010

  49. [49]

    Vaswani, N

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin. Attention is all you need. Adv. Neural Inf. Process. Syst. , 30:5998–6008, 2017

    work page 2017

  50. [50]

    Z. Zhao, Z. Luo, J. Sj¨ olund, and T. B. Sch¨ on. Conditional sampling within generative diffusion models. arXiv:2409.09650, 2024. Appendix A. Implementation details A.1. Networks and training. The spaces NNΘ,p, p = 1 or p = d, where we define our models consist of fully connected feed-forward neural networks with three hidden layers, one input layer, and...

    work page arXiv 2024