arXiv:2605.28858v1 Announce Type: cross 
Abstract: This work presents an end-to-end strategy for solving inverse problems constrained by Partial Differential Equations within a fully differentiable Machine Learning framework. The proposed formulation provides a unified and user-friendly methodology applicable to a wide range of problems, from data assimilation to closure modeling. Our approach combines a baseline differentiable PDE solver, which predicts the state w from the nonlinear system $R(w) = 0$, with a generic additive, parametrized, and differentiable correction $f_\phi(w)$, with trainable parameters $\phi$. We show how to optimize phi within a fully differentiable Python workflow by reformulating the PDE as an implicit layer, enabling its integration into arbitrary objective functions, while leveraging PyTorch's automatic differentiation graph. The method is demonstrated on the Reynolds-Averaged Navier-Stokes equations for compressible flows, where the closure term, or a portion of it, is modeled using trainable parameters or a Neural Network. The first application considers the 2D NASA Wall-Mounted Hump test case, where a production-term parameter is optimized against time-averaged LES data. A second application is carried out on the VKI LS-59 turbine blade, where the Spalart-Allmaras eddy viscosity field is reconstructed through the optimization of a trainable spatial field. A dataset is generated starting from the VKI LS-59 turbine blade geometry using the differentiable BROADCAST solver with the Spalart-Allmaras turbulence model. The results highlight the flexibility of the framework, showing its applicability beyond turbulence modeling to a broader class of physics-informed PDE-constrained problems with data-driven components.

قدمت دراسة حديثة استراتيجية شاملة لحل المشكلات العكسية المقيدة بمعادلات تفاضلية جزئية (PDE) ضمن إطار تعلم آلي قابل للاشتقاق بالكامل، باستخدام PyTorch. يجمع هذا النهج بين حل PDE القابل للاشتقاق الأساسي وتصحيح معلم، مما يسمح بالتحسين ضمن سير عمل بايثون قابل للاشتقاق بالكامل. يتم توضيح الطريقة من خلال تطبيقها على معادلات نافير-ستوكس المتوسطة بواسطة رينولدز لتدفقات قابلة للضغط.

Un estudio reciente ha presentado una estrategia integral para resolver problemas inversos restringidos por ecuaciones en derivadas parciales (EDP) dentro de un marco de aprendizaje automático completamente diferenciable, utilizando PyTorch. Este enfoque combina un solucionador de EDP diferenciable básico con una corrección parametrizada, permitiendo la optimización dentro de un flujo de trabajo de Python completamente diferenciable. El método se ejemplifica a través de su aplicación a las ecuaciones de Navier-Stokes promediadas por Reynolds para flujos compresibles.

Une étude récente a introduit une stratégie de bout en bout pour résoudre des problèmes inverses contraints par des équations aux dérivées partielles (EDP) dans un cadre d'apprentissage automatique entièrement différentiable, utilisant PyTorch. Cette approche combine un solveur EDP différentiable de base avec une correction paramétrée, permettant l'optimisation dans un flux de travail Python entièrement différentiable. La méthode est illustrée par son application aux équations de Navier-Stokes moyennées par Reynolds pour des flux compressibles.

A recent study has introduced an end-to-end strategy for solving inverse problems constrained by Partial Differential Equations (PDEs) within a fully differentiable Machine Learning framework, utilizing PyTorch. This approach combines a baseline differentiable PDE solver with a parametrized correction, allowing for optimization within a fully differentiable Python workflow. The method is exemplified through its application to the Reynolds-Averaged Navier-Stokes equations for compressible flows.