Abstract
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
$ \boldsymbol{w} $
w
from the nonlinear system
R
w
=
0
$ \boldsymbol{R}\left(\boldsymbol{w}\right)=0 $
Rw equals 0
, with a generic additive, parametrized, and differentiable correction
f
ϑ
w
$ {\boldsymbol{f}}_{\boldsymbol{\vartheta}}\left(\boldsymbol{w}\right) $
f theta w
, with trainable parameters
ϑ
$ \boldsymbol{\vartheta} $
theta
. We show how to optimize
ϑ
$ \boldsymbol{\vartheta} $
theta
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.