Non-Symmetric Information Operators in GMRFs for PDE Inference

Yumeng Shi, Concha Bielza, Pedro Larrañaga
Proceedings of the 13th International Conference on Probabilistic Graphical Models, PMLR 343:835-849, 2026.

Abstract

Probabilistic approaches to partial differential equations (PDEs) cast the solution as a Bayesian inference problem, in which the unknown field is inferred from a prior distribution and a likelihood derived from the governing equations. When the prior is Gaussian-Markov, the solution admits a Gaussian Markov Random Field (GMRF) representation with a sparse precision matrix, enabling efficient inference and marginal uncertainty estimates. In this framework, physical constraints enter through information operators that define the likelihood. Standard formulations rely on symmetric operators, which encode a non-directional flow of information. In advection-dominated regimes, this mismatch produces spurious oscillations and unreliable uncertainty estimates. In this work we introduce a non-symmetric information operator that embeds a physically motivated directional preference into the likelihood. Although the operator is non-symmetric, the resulting posterior precision matrix remains symmetric and positive definite, preserving full compatibility with standard sparse GMRF inference. We evaluate the method on the viscous Burgers’ equation across diffusion- to advection-dominated regimes. The non-symmetric operator suppresses spurious oscillations and matches finite-element accuracy. Its posterior uncertainty remains reliable in advection-dominated regimes where the symmetric operator becomes overconfident.

Cite this Paper

BibTeX
@InProceedings{pmlr-v343-shi26a, title = {Non-Symmetric Information Operators in GMRFs for {PDE} Inference}, author = {Shi, Yumeng and Bielza, Concha and Larra\~naga, Pedro}, booktitle = {Proceedings of the 13th International Conference on Probabilistic Graphical Models}, pages = {835--849}, year = {2026}, editor = {Camps-Valls, Gustau and Leonelli, Manuele and Varando, Gherardo}, volume = {343}, series = {Proceedings of Machine Learning Research}, month = {09--11 Sep}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v343/main/assets/shi26a/shi26a.pdf}, url = {https://proceedings.mlr.press/v343/shi26a.html}, abstract = {Probabilistic approaches to partial differential equations (PDEs) cast the solution as a Bayesian inference problem, in which the unknown field is inferred from a prior distribution and a likelihood derived from the governing equations. When the prior is Gaussian-Markov, the solution admits a Gaussian Markov Random Field (GMRF) representation with a sparse precision matrix, enabling efficient inference and marginal uncertainty estimates. In this framework, physical constraints enter through information operators that define the likelihood. Standard formulations rely on symmetric operators, which encode a non-directional flow of information. In advection-dominated regimes, this mismatch produces spurious oscillations and unreliable uncertainty estimates. In this work we introduce a non-symmetric information operator that embeds a physically motivated directional preference into the likelihood. Although the operator is non-symmetric, the resulting posterior precision matrix remains symmetric and positive definite, preserving full compatibility with standard sparse GMRF inference. We evaluate the method on the viscous Burgers’ equation across diffusion- to advection-dominated regimes. The non-symmetric operator suppresses spurious oscillations and matches finite-element accuracy. Its posterior uncertainty remains reliable in advection-dominated regimes where the symmetric operator becomes overconfident.} }
Endnote
%0 Conference Paper %T Non-Symmetric Information Operators in GMRFs for PDE Inference %A Yumeng Shi %A Concha Bielza %A Pedro Larrañaga %B Proceedings of the 13th International Conference on Probabilistic Graphical Models %C Proceedings of Machine Learning Research %D 2026 %E Gustau Camps-Valls %E Manuele Leonelli %E Gherardo Varando %F pmlr-v343-shi26a %I PMLR %P 835--849 %U https://proceedings.mlr.press/v343/shi26a.html %V 343 %X Probabilistic approaches to partial differential equations (PDEs) cast the solution as a Bayesian inference problem, in which the unknown field is inferred from a prior distribution and a likelihood derived from the governing equations. When the prior is Gaussian-Markov, the solution admits a Gaussian Markov Random Field (GMRF) representation with a sparse precision matrix, enabling efficient inference and marginal uncertainty estimates. In this framework, physical constraints enter through information operators that define the likelihood. Standard formulations rely on symmetric operators, which encode a non-directional flow of information. In advection-dominated regimes, this mismatch produces spurious oscillations and unreliable uncertainty estimates. In this work we introduce a non-symmetric information operator that embeds a physically motivated directional preference into the likelihood. Although the operator is non-symmetric, the resulting posterior precision matrix remains symmetric and positive definite, preserving full compatibility with standard sparse GMRF inference. We evaluate the method on the viscous Burgers’ equation across diffusion- to advection-dominated regimes. The non-symmetric operator suppresses spurious oscillations and matches finite-element accuracy. Its posterior uncertainty remains reliable in advection-dominated regimes where the symmetric operator becomes overconfident.
APA
Shi, Y., Bielza, C. & Larrañaga, P.. (2026). Non-Symmetric Information Operators in GMRFs for PDE Inference. Proceedings of the 13th International Conference on Probabilistic Graphical Models, in Proceedings of Machine Learning Research 343:835-849 Available from https://proceedings.mlr.press/v343/shi26a.html.

Related Material