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arxiv: 2605.31127 · v1 · pith:5A3UHWTInew · submitted 2026-05-29 · 💻 cs.LG · cs.NA· math.NA

Scalable Bayesian Inference for Nonlinear Conservation Laws

Tim Weiland , Philipp Hennig This is my paper

Pith reviewed 2026-06-28 23:16 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NA
keywords Bayesian inferencenonlinear conservation lawsGaussian processesuncertainty quantificationinverse problemssparse approximationsnumerical methodsphysics-informed inference
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The pith

A method integrates sparse Gaussian process approximations with classical conservative discretizations to enable scalable Bayesian inference for nonlinear conservation laws.

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

The paper establishes a numerically conservative Bayesian approach for simulating nonlinear conservation laws under uncertainty by framing classical solvers as inference under Gaussian process priors and applying sparse approximations for scalability. This produces forward simulations that match classical solver accuracy while adding structured uncertainty quantification, and it solves inverse problems by recovering full posteriors over nonparametric source fields in seconds. A sympathetic reader would care because many engineering and scientific systems governed by conservation laws face sparse or noisy data, where standard numerical methods ignore uncertainty and deep learning approaches often sacrifice conservation or speed. The central object is the integration of sparse GPs with finite-volume style discretizations that preserves conservation properties even for nonlinear cases.

Core claim

We develop a novel numerically conservative method for uncertainty-aware simulations of nonlinear conservation laws. We use recent sparse approximation techniques to scale up to large-scale forward and inverse problems. For forward simulation, we inherit the accuracy of classical solvers while providing structured uncertainty quantification. On inverse problems, we recover posteriors over nonparametric source fields in seconds -- outperforming neural baselines that take minutes to produce a less accurate point estimate.

What carries the argument

Sparse Gaussian process approximations integrated with classical conservative discretizations under Gaussian process priors, preserving numerical conservation while enabling uncertainty quantification.

If this is right

  • Forward simulations match the accuracy of classical solvers while adding structured uncertainty quantification.
  • Inverse problems recover posteriors over nonparametric source fields in seconds rather than minutes.
  • The approach scales to large forward and inverse problems where prior methods struggle.
  • Neural baselines are outperformed in both speed and accuracy for point estimates on inverse tasks.

Where Pith is reading between the lines

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

  • The same integration pattern could be tested on other classes of PDEs if conservation is not the primary constraint.
  • Structured uncertainty from this method might directly support risk-aware control in engineering systems with conservation constraints.
  • Real-world validation would involve applying the method to measurement data from actual physical experiments rather than synthetic benchmarks.

Load-bearing premise

Sparse Gaussian process approximations can be integrated with classical conservative discretizations so that numerical conservation and accuracy properties hold for nonlinear problems.

What would settle it

A closed-system test case where the computed solutions violate a conservation law, such as total mass or momentum not remaining constant within machine precision over time.

Figures

Figures reproduced from arXiv: 2605.31127 by Philipp Hennig, Tim Weiland.

Figure 1
Figure 1. Figure 1: Source identification from sparse sensor measurements. Top row: concentration field. Bottom row: inferred source field. Our GP-FVM method (right) recovers the source locations with quantified uncertainty, while a PINN baseline produces a point estimate with artifacts. Our method solves this nonparametric problem in seconds; the PINN requires minutes. numerical scheme has remained an open problem. Recent wo… view at source ↗
Figure 2
Figure 2. Figure 2: Posterior samples adapt to observations. Each column shows a posterior sample of the source field using the same random seed. Top row: 12 observations (◦). Bottom row: 15 observations (3 additional shown as ♦). The added observations correctly inform the posterior about the weakness of the source in the top left corner, causing all samples to shift mass away from that region. s(x, y). This problem is ill-p… view at source ↗
Figure 4
Figure 4. Figure 4: Kernel smoothness dictates convergence order. • Matern-3/2, ´ ■ Matern-5/2, ´ ♦ Matern-7/2. Dashed lines show ´ theoretical rates. (a) Dense GP-FVM achieves higher-order con￾vergence. (b) Sparse approximation (ρ = 5, 16% fill) preserves convergence for Matern-3/2; smoother kernels plateau at fine reso- ´ lutions. FVM, a collocation-based sparse GP approach as in Chen et al. (2025), and our GP-FVM in both f… view at source ↗
Figure 5
Figure 5. Figure 5: Nonlinear shallow water simulation. Evolution of the posterior mean over the surface elevation η over a domain with spatially varying bathymetry (bottom surface). The method handles this system of three coupled conservation laws, demonstrating scalability to complex nonlinear problems. 6. Discussion We frame the finite volume method as nonlinear condition￾ing under a structured GP prior. An extension of Ve… view at source ↗
Figure 6
Figure 6. Figure 6: Ordering comparison for sparse Cholesky approximation. (a) Fill-in percentage vs sparsity parameter ρ. (b) KL divergence vs ρ. (c) Pareto frontier showing KL divergence vs fill-in. The “integrals first” ordering (red) achieves lower error at every sparsity level, demonstrating Pareto dominance over “evaluations first” (blue). Ordering algorithm. We order functional blocks from finest to coarsest (derivativ… view at source ↗
Figure 7
Figure 7. Figure 7: Exact precision Cholesky factor for 2D mixed functionals. Log-magnitude of entries in the lower triangular factor L for a 12 × 12 grid (409 × 409 matrix). (a) Integrals-coarsest ordering places integrals in the bottom-right block. (b) Evaluations-coarsest ordering. Both show rapid decay of remote entries (dark blue regions), justifying sparse approximation. Dashed lines indicate block boundaries. Our Julia… view at source ↗
Figure 8
Figure 8. Figure 8: Calibration plot over 200 random two-source instances (31 × 31 grid). Empirical coverage vs. nominal coverage level. The concentration field (blue) tracks the diagonal closely with mild deviations at the extremes; the source field (red) is broadly conservative and matches the diagonal near 95%. x 0.0 0.5 1.0 y 0.0 0.5 1.0 Source x 0.0 0.5 1.0 Concentration Empirical coverage at 95% nomina 0.80 0.85 0.90 0.… view at source ↗
Figure 9
Figure 9. Figure 9: Spatial coverage map at 95% nominal level, averaged over 200 instances. Values near 0.95 (white) indicate well-calibrated regions. The source field is slightly overconfident in the interior where sources are active. Output scale calibration. The GP prior’s output scale σ 2 controls the overall magnitude of the posterior uncertainty. We calibrate it via a two-pass procedure: (i) run 10 calibration instances… view at source ↗
Figure 10
Figure 10. Figure 10: Wall-clock time vs. total degrees of freedom Ns for the source identification problem. The dashed line shows the theoretical O(N 3/2 s ) scaling from nested-dissection sparse Cholesky. The total time (black) tracks this reference closely. Discussion. The remaining miscalibration has two distinct sources: 1. Prior–truth mismatch in the source field. The stationary Matern prior spreads uncertainty uniformly… view at source ↗
Figure 11
Figure 11. Figure 11: Nonlinear inverse problem: 2D Burgers source identification. Top: ground truth u(x, y, t). Middle: GP-FVM posterior mean. Bottom: posterior standard deviation; white dots indicate observation locations. The method correctly captures the nonlinear advection to the upper right while jointly inferring the unknown source. F. Gauss–Newton Ablation The sequential (EKF-style) solver of Section 3.4 performs a sin… view at source ↗
Figure 12
Figure 12. Figure 12: Inferred source field. Left to right: ground truth s(x, y), posterior mean softplus(ˆg), absolute error, and posterior std in g-space. Both source bumps are identified; the dominant source (amplitude ≈ 3.5) is well-recovered. Gauss-Newton iterations per timestep 1 2 5 10 Relative L2 error (final ti me) 0.00 0.05 0.10 0.15 0.20 [PITH_FULL_IMAGE:figures/full_fig_p022_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Gauss–Newton ablation for the EKF sequential solver on 1D Burgers (N = 100, 5 Gaussian ICs). Additional iterations beyond 1 provide no accuracy improvement. contribution that vanishes as ∆t → 0 and a diagonal mismatch floor controlled by the spatial prior’s conditioning. G.1. Setup and Notation Definition G.1 (Approximating family). Let Qs ≻ 0 be the sparse spatial prior precision. The approximating famil… view at source ↗
read the original abstract

Nonlinear conservation laws are at the heart of many of the most important dynamical systems in science and engineering. In practical applications, such systems are often subject to various sources of uncertainty, e.g. due to sparse or noisy measurements. Inferring physical quantities and fields of interest then becomes an ill-posed problem which both classical numerical methods and modern deep learning-based methods struggle to treat appropriately. Recent work has framed classical numerical methods as Bayesian inference under Gaussian process priors, resulting in a physics-aware treatment of uncertainties. Following this line of work, we develop a novel numerically conservative method for uncertainty-aware simulations of nonlinear conservation laws. We use recent sparse approximation techniques to scale up to large-scale forward and inverse problems. For forward simulation, we inherit the accuracy of classical solvers while providing structured uncertainty quantification. On inverse problems, we recover posteriors over nonparametric source fields in seconds -- outperforming neural baselines that take minutes to produce a less accurate point estimate.

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

1 major / 0 minor

Summary. The paper develops a novel numerically conservative Bayesian method for uncertainty-aware simulation of nonlinear conservation laws. It integrates sparse Gaussian process approximations with classical conservative discretizations to scale both forward problems (inheriting classical accuracy plus structured UQ) and inverse problems (recovering posteriors over nonparametric source fields in seconds, outperforming neural baselines that produce less accurate point estimates after minutes of computation).

Significance. If the claimed preservation of discrete conservation and accuracy under sparsification holds, the work would meaningfully advance physics-informed probabilistic inference for PDEs by enabling scalable, uncertainty-aware computations on problems where both classical solvers and neural methods currently fall short.

major comments (1)
  1. [Abstract and method description] The central claim that sparse GP approximations can be fused with conservative discretizations while exactly preserving the telescoping property (and thus discrete conservation) for nonlinear fluxes is load-bearing for both the forward and inverse results. The abstract asserts inheritance of accuracy and conservation, yet the integration step for nonlinear update maps is not guaranteed to survive sparsification (inducing points or variational approximations) without a modified stencil or re-derived scheme that retains the global sum. This must be shown explicitly, e.g., via a conservation-error table or proof in the method section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for emphasizing the need to explicitly verify preservation of discrete conservation. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and method description] The central claim that sparse GP approximations can be fused with conservative discretizations while exactly preserving the telescoping property (and thus discrete conservation) for nonlinear fluxes is load-bearing for both the forward and inverse results. The abstract asserts inheritance of accuracy and conservation, yet the integration step for nonlinear update maps is not guaranteed to survive sparsification (inducing points or variational approximations) without a modified stencil or re-derived scheme that retains the global sum. This must be shown explicitly, e.g., via a conservation-error table or proof in the method section.

    Authors: We agree that an explicit demonstration is required. The sparse variational approximation is constructed so that the inducing-point posterior is optimized under the same conservative finite-volume likelihood used in the dense case; the nonlinear flux evaluations therefore continue to telescope exactly on the grid. Nevertheless, to address the concern directly we will add (i) a short proof in Section 3 showing that the variational free-energy objective does not alter the global conservation identity for any nonlinear flux, and (ii) a conservation-error table in the numerical experiments that reports the discrete global-sum deviation (to machine precision) for both the full GP and several sparsification levels. revision: yes

Circularity Check

0 steps flagged

No circularity; novel conservative sparse-GP method builds on but does not reduce to prior framing

full rationale

The paper states it follows 'recent work' framing numerical methods as Bayesian inference under GP priors, then develops a novel numerically conservative method using sparse approximations. No quoted equations or steps show a prediction reducing to a fitted input by construction, a self-definitional loop, or a load-bearing self-citation whose validity is internal to this manuscript. The conservation and scaling claims are presented as new contributions whose validity is independent of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the method appears to rest on standard Gaussian process and sparse approximation machinery from prior literature.

pith-pipeline@v0.9.1-grok · 5687 in / 1215 out tokens · 31607 ms · 2026-06-28T23:16:40.252744+00:00 · methodology

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Reference graph

Works this paper leans on

31 extracted references · 18 canonical work pages

  1. [1]

    Dynamics of Fluids in Porous Media

    Bear, J. Dynamics of Fluids in Porous Media . Courier Corporation, January 1988. ISBN 978-0-486-65675-5

    1988

  2. [2]

    A Dictionary of Closed-Form Kernel Mean Embeddings

    Briol, F.-X., Karvonen, T., Gessner, A., and Mahsereci, M. A Dictionary of Closed-Form Kernel Mean Embeddings . In Proceedings of the First International Conference on Probabilistic Numerics , pp.\ 84--94. PMLR, August 2025

    2025

  3. [3]

    A., Hager, W

    Chen, Y., Davis, T. A., Hager, W. W., and Rajamanickam, S. Algorithm 887: CHOLMOD , Supernodal Sparse Cholesky Factorization and Update / Downdate . ACM Transactions on Mathematical Software, 35 0 (3): 0 1--14, October 2008. ISSN 0098-3500, 1557-7295. doi:10.1145/1391989.1391995

    work page doi:10.1145/1391989.1391995 2008

  4. [4]

    Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

    Chen, Y., Owhadi, H., and Sch \"a fer, F. Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes. Mathematics of Computation, 94 0 (353): 0 1235--1280, May 2025. ISSN 0025-5718, 1088-6842. doi:10.1090/mcom/3992

    work page doi:10.1090/mcom/3992 2025

  5. [5]

    Bayesian Probabilistic Numerical Methods

    Cockayne, J. Bayesian Probabilistic Numerical Methods . PhD thesis, University of Warwick, July 2019

    2019

  6. [6]

    Finite Volume Methods

    Eymard, R., Gallou \"e t, T., and Herbin, R. Finite Volume Methods . In Lions, J. L. and Ciarlet, P. (eds.), Solution of Equation in R n ( Part 3), Techniques of Scientific Computing ( Part 3) , volume 7 of Handbook of Numerical Analysis , pp.\ 713--1020. Elsevier, 2000. doi:10.1016/S1570-8659(00)07005-8

    work page doi:10.1016/s1570-8659(00)07005-8 2000

  7. [7]

    A., and Kersting, H

    Hennig, P., Osborne, M. A., and Kersting, H. P. Probabilistic Numerics : Computation as Machine Learning . Cambridge University Press, Cambridge, 2022. ISBN 978-1-107-16344-7. doi:10.1017/9781316681411

    work page doi:10.1017/9781316681411 2022

  8. [8]

    and Dalle, G

    Hill, A. and Dalle, G. Sparser, Better , Faster , Stronger : Sparsity Detection for Efficient Automatic Differentiation , June 2025

    2025

  9. [9]

    and Koutsourelakis, P.-S

    Kaltenbach, S. and Koutsourelakis, P.-S. Incorporating Physical Constraints in a Deep Probabilistic Machine Learning Framework for Coarse-Graining Dynamical Systems . Journal of Computational Physics, 419: 0 109673, 2020. ISSN 0021-9991. doi:10.1016/j.jcp.2020.109673

    work page doi:10.1016/j.jcp.2020.109673 2020

  10. [10]

    and Guinness, J

    Katzfuss, M. and Guinness, J. A general framework for Vecchia approximations of Gaussian processes. Statistical Science, 36 0 (1), February 2021. ISSN 0883-4237. doi:10.1214/19-STS755

    work page doi:10.1214/19-sts755 2021

  11. [11]

    Neural operator: Learning maps between function spaces

    Kovachki, N., Li, Z., Liu, B., Azizzadenesheli, K., Bhattacharya, K., Stuart, A., and Anandkumar, A. Neural operator: Learning maps between function spaces. Journal of Machine Learning Research, 24 0 (89): 0 4061--4157, 2023. ISSN 1533-7928

    2023

  12. [12]

    Probabilistic ODE Solutions in Millions of Dimensions

    Kr \"a mer, N., Bosch, N., Schmidt, J., and Hennig, P. Probabilistic ODE Solutions in Millions of Dimensions . In Proceedings of the 39th International Conference on Machine Learning , volume 162 of Proceedings of Machine Learning Research, pp.\ 11634--11649. PMLR, 2022

    2022

  13. [13]

    Finite-volume schemes for shallow-water equations

    Kurganov, A. Finite-volume schemes for shallow-water equations. Acta Numerica, 27: 0 289--351, May 2018. ISSN 0962-4929, 1474-0508. doi:10.1017/S0962492918000028

    work page doi:10.1017/s0962492918000028 2018

  14. [14]

    Lauritzen, P., Jablonowski, C., Taylor, M., and Nair, R. (eds.). Numerical Techniques for Global Atmospheric Models , volume 80 of Lecture Notes in Computational Science and Engineering . Springer, Berlin, Heidelberg, 2011. ISBN 978-3-642-11639-1 978-3-642-11640-7. doi:10.1007/978-3-642-11640-7

    work page doi:10.1007/978-3-642-11640-7 2011

  15. [15]

    C., Lu, J., Ying, L., and E, W

    Lin, L., Yang, C., Meza, J. C., Lu, J., Ying, L., and E, W. SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix . ACM Trans. Math. Softw., 37 0 (4): 0 40:1--40:19, February 2011. ISSN 0098-3500. doi:10.1145/1916461.1916464

    work page doi:10.1145/1916461.1916464 2011

  16. [16]

    J., Rose, D

    Lipton, R. J., Rose, D. J., and Tarjan, R. E. Generalized Nested Dissection . SIAM Journal on Numerical Analysis, 16 0 (2): 0 346--358, 1979. doi:10.1137/0716027

    work page doi:10.1137/0716027 1979

  17. [17]

    Bayes -- Hermite quadrature

    O'Hagan, A. Bayes -- Hermite quadrature. Journal of Statistical Planning and Inference, 29 0 (3): 0 245--260, November 1991. ISSN 0378-3758. doi:10.1016/0378-3758(91)90002-V

    work page doi:10.1016/0378-3758(91)90002-v 1991

  18. [18]

    Physics- Informed Gaussian Process Regression Generalizes Linear PDE Solvers , April 2024

    Pf \"o rtner, M., Steinwart, I., Hennig, P., and Wenger, J. Physics- Informed Gaussian Process Regression Generalizes Linear PDE Solvers , April 2024

    2024

  19. [19]

    T., and Kohandel, M

    Podina, L., Rad, M. T., and Kohandel, M. Conformalized Physics-Informed Neural Networks , May 2024

    2024

  20. [20]

    Raissi, M., Perdikaris, P., and Karniadakis, G. E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378: 0 686--707, February 2019. ISSN 0021-9991. doi:10.1016/j.jcp.2018.10.045

    work page doi:10.1016/j.jcp.2018.10.045 2019

  21. [21]

    Sparse Cholesky Factorization by Kullback--Leibler Minimization

    Sch \"a fer, F., Katzfuss, M., and Owhadi, H. Sparse Cholesky Factorization by Kullback--Leibler Minimization . SIAM J. Sci. Comput., 43 0 (3): 0 A2019--A2046, January 2021. ISSN 1064-8275. doi:10.1137/20M1336254

    work page doi:10.1137/20m1336254 2021

  22. [22]

    A Probabilistic State Space Model for Joint Inference from Differential Equations and Data

    Schmidt, J., Kr \"a mer, N., and Hennig, P. A Probabilistic State Space Model for Joint Inference from Differential Equations and Data . In Advances in Neural Information Processing Systems , volume 34, pp.\ 12374--12385. Curran Associates, Inc., 2021

    2021

  23. [23]

    A probabilistic model for the numerical solution of initial value problems

    Schober, M., S \"a rkk \"a , S., and Hennig, P. A probabilistic model for the numerical solution of initial value problems. Statistics and Computing, 29 0 (1): 0 99--122, January 2019. ISSN 1573-1375. doi:10.1007/s11222-017-9798-7

    work page doi:10.1007/s11222-017-9798-7 2019

  24. [24]

    Stein, M. L. When does the screening effect hold? The Annals of Statistics, 39 0 (6), December 2011. ISSN 0090-5364. doi:10.1214/11-AOS909

    work page doi:10.1214/11-aos909 2011

  25. [25]

    Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: A new perspective

    Tronarp, F., Kersting, H., S \"a rkk \"a , S., and Hennig, P. Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: A new perspective. Statistics and Computing, 29 0 (6): 0 1297--1315, November 2019. ISSN 1573-1375. doi:10.1007/s11222-019-09900-1

    work page doi:10.1007/s11222-019-09900-1 2019

  26. [26]

    B., Rizzi, A., Darracq, D., and Hirschel, E

    Vos, J. B., Rizzi, A., Darracq, D., and Hirschel, E. H. Navier -- Stokes solvers in European aircraft design. Progress in Aerospace Sciences, 38 0 (8): 0 601--697, November 2002. ISSN 0376-0421. doi:10.1016/S0376-0421(02)00050-7

    work page doi:10.1016/s0376-0421(02)00050-7 2002

  27. [27]

    An Expert 's Guide to Training Physics-informed Neural Networks , August 2023

    Wang, S., Sankaran, S., Wang, H., and Perdikaris, P. An Expert 's Guide to Training Physics-informed Neural Networks , August 2023

    2023

  28. [28]

    GaussianMarkovRandomFields.jl , v0.9.1

    Weiland, T. GaussianMarkovRandomFields.jl , v0.9.1. Zenodo, December 2025

    2025

  29. [29]

    Scaling up Probabilistic PDE Simulators with Structured Volumetric Information , June 2024

    Weiland, T., Pf \"o rtner, M., and Hennig, P. Scaling up Probabilistic PDE Simulators with Structured Volumetric Information , June 2024

    2024

  30. [30]

    Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields

    Weiland, T., Pf \"o rtner, M., and Hennig, P. Flexible and Efficient Probabilistic PDE Solvers through Gaussian Markov Random Fields . In Proceedings of The 28th International Conference on Artificial Intelligence and Statistics , pp.\ 2746--2754. PMLR, April 2025

    2025

  31. [31]

    Scattered Data Approximation

    Wendland, H. Scattered Data Approximation . Cambridge Monographs on Applied and Computational Mathematics . Cambridge University Press, Cambridge, 2004. ISBN 978-0-521-84335-5. doi:10.1017/CBO9780511617539

    work page doi:10.1017/cbo9780511617539 2004