An End-to-End PyTorch Interface for Differentiable PDE Solvers: A RANS Model-Correction Study
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.
Versions
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BibTeX
@article{saverio2026endtoend,
title = {{An End-to-End PyTorch Interface for Differentiable PDE Solvers: A RANS Model-Correction Study}},
author = {Luca Saverio and Michele Alessandro Bucci and Gianmarco Farro and Cédric Content and Denis Sipp},
journal = {Data-Centric Engineering},
year = {2026},
month = may,
doi = {10.1017/dce.2026.10066},
eprint = {2605.28858},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05621488v2},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - JOUR
TI - An End-to-End PyTorch Interface for Differentiable PDE Solvers: A RANS Model-Correction Study
AU - Luca Saverio
AU - Michele Alessandro Bucci
AU - Gianmarco Farro
AU - Cédric Content
AU - Denis Sipp
PY - 2026
DA - 2026/05/19
JO - Data-Centric Engineering
DO - 10.1017/dce.2026.10066
UR - https://hal.science/hal-05621488v2
AB - 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.
SN - 2632-6736
ER -