arXiv:2512.12749v2 Announce Type: replace-cross 
Abstract: Learning probabilistic surrogates for partial differential equations remains challenging in data-scarce regimes: neural operators require large amounts of high-fidelity data, while generative approaches typically sacrifice resolution invariance. We formulate flow matching in an infinite-dimensional function space to learn a probabilistic transport that maps low-fidelity approximations to the manifold of high-fidelity PDE solutions via learned residual corrections. We develop a conditional neural operator architecture based on feature-wise linear modulation for flow matching vector fields directly in function space, enabling inference at arbitrary spatial resolutions without retraining. To improve stability and representational control of the induced neural ODE, we parameterize the flow vector field as a sum of a linear operator and a nonlinear operator, combining lightweight linear components with a conditioned Fourier neural operator for expressive, input-dependent dynamics. We then formulate a residual-augmented learning strategy where the flow model learns probabilistic corrections from inexpensive low-fidelity surrogates to high-fidelity solutions, rather than learning the full solution mapping from scratch. Finally, we derive tractable training objectives that extend conditional flow matching to the operator setting with input-function-dependent couplings. To demonstrate the effectiveness of our approach, we present numerical experiments on a range of PDEs, including the 1D advection and Burgers' equation, and a 2D Darcy flow problem for flow through a porous medium. We show that the proposed method can accurately learn solution operators across different resolutions and fidelities and produces uncertainty estimates that appropriately reflect model confidence, even when trained on limited high-fidelity data.

تم تقديم إطار جديد لتعلم البدائل الاحتمالية للمعادلات التفاضلية الجزئية (PDE)، مما يعالج التحديات في البيئات التي تعاني من نقص البيانات. يستخدم هذا الإطار مطابقة التدفق في فضاء دالة ذو أبعاد غير محدودة لتعزيز تحويل التقريبات ذات الدقة المنخفضة إلى حلول PDE عالية الدقة من خلال تصحيحات متبقية متعلمة.

Se ha introducido un nuevo marco para el aprendizaje de sustitutos probabilísticos para ecuaciones en derivadas parciales (EDP), abordando los desafíos en entornos con escasez de datos. Este marco utiliza el emparejamiento de flujo en un espacio funcional de dimensión infinita para mejorar la asignación de aproximaciones de baja fidelidad a soluciones de EDP de alta fidelidad mediante correcciones residuales aprendidas.

Un nouveau cadre pour l'apprentissage de substituts probabilistes pour les équations aux dérivées partielles (EDP) a été introduit, abordant les défis dans des environnements où les données sont rares. Ce cadre utilise l'appariement de flux dans un espace fonctionnel de dimension infinie pour améliorer la cartographie des approximations de faible fidélité vers des solutions EDP de haute fidélité grâce à des corrections résiduelles apprises.

A new framework for learning probabilistic surrogates for partial differential equations (PDEs) has been introduced, addressing challenges in data-scarce environments. This framework utilizes flow matching in infinite-dimensional function space to enhance the mapping of low-fidelity approximations to high-fidelity PDE solutions through learned residual corrections.

Flow matching Operators for Residual-Augmented Probabilistic Learning of Partial Differential Equations

Was this article worth reading? Share it

LucidQuery AI

Airparser

Metaflow AI

Dynamiq

GPTHumanizer

Mapfit

Ready to build your own newsroom?