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 $f_φ(w)$, with trainable parameters $φ$. 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.

翻译：本文提出了一种在全可微机器学习框架内求解偏微分方程约束反问题的端到端策略。该方案提供了一种统一且用户友好的方法论，可广泛应用于从数据同化到闭合建模的各类问题。我们的方法结合了基线可微PDE求解器（该求解器从非线性系统$R(w)=0$中预测状态w）与泛型可加、参数化且可微的修正项$f_φ(w)$（其中φ为可训练参数）。通过将PDE重构为隐式层，我们展示了如何在完全可微的Python工作流中优化φ参数，使其能够集成到任意目标函数中，同时充分利用PyTorch的自动微分计算图。该方法在可压缩流的雷诺平均纳维-斯托克斯方程上得到验证：其闭合项（或部分闭合项）通过可训练参数或神经网络进行建模。第一个应用案例针对二维NASA壁挂凸包测试问题，通过优化生产项参数使其与时间平均LES数据匹配。第二个应用在VKI LS-59涡轮叶片上展开，通过优化可训练空间场重构了Spalart-Allmaras涡粘性场。基于VKI LS-59涡轮叶片几何，利用可微BROADCAST求解器结合Spalart-Allmaras湍流模型生成数据集。结果凸显了该框架的灵活性，证明其可超越湍流建模范畴，适用于更广泛的基于物理信息的PDE约束问题（含数据驱动组件）。