Automatic Differentiation with Constraints in Forward and Inverse Problems

Nikolay Yavich; Alexander Ryabov; Viacheslav Naumov; Sayan Ranu; N. M. Anoop Krishnan; Evgeny Burnaev; Vladimir Vanovskiy

cs.AIOct 20, 2026

Automatic Differentiation with Constraints in Forward and Inverse Problems

Authors: Nikolay Yavich, Alexander Ryabov, Viacheslav Naumov, Sayan Ranu, N. M. Anoop Krishnan, Evgeny Burnaev, Vladimir Vanovskiy

Organizations: Artificial Intelligence Center, Skolkovo Institute of Science and Technology, 121205 Moscow, Russia · Yardi School of Artificial Intelligence, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, India 110016 · Learnable Intelligence Group, Autonomous Non-Profit Organization Artificial Intelligence Research Institute (AIRI), 105064 Moscow, Russia

Abstract

Approximate solutions of partial differential equations (PDEs) on unstructured grids can be represented on a graph. Numerical optimization involving stiff evolutionary and static PDEs often requires solving a large system of equations, which can be treated as a constraint. In this work, we present a fully differentiable graph-based implicit solver that incorporates this constraint. We evaluate its performance across four key tasks: a) linear and nonlinear forward modeling of evolutionary and static problems, b) solving inverse problems to estimate equation coefficients, c) solving inverse problems to estimate the right-hand side, and d) learnable graph coarsening to accelerate forward modeling. The stability and differentiability of our solver enable significant reductions in the complexity of forward modeling while efficiently addressing related inverse problems. These capabilities make it a powerful tool for different physics and engineering applications, in particular for geophysics, where inverse problems are commonly solved in order to reduce the model uncertainty.

Figures & tables

Figure 1: Differentiable modeling pipeline developed in this work combines graph neural networks, the finite-volume method, implicit time-stepping and can be applied to optimization in forward and inverse problems.

Figure 2: Computational grid coarsening for the linear parabolic equation, ( 1 ). a) Original grid, color indicates permeability, source marked with blued diamond, and two measurement points (1 and 2) marked as black diamonds; b) coarsened grid; c) data at two measurement points, modelled with the original and coarsened grids; d) loss function during optimization.

Figure 3: Computational grid coarsening for the nonlinear parabolic equation, ( 17 ). a) Original grid, color indicates permeability, source marked with blued diamond, and two measurement points (1 and 2) marked as black diamonds; b) coarsened grid; c) data at two measurement points, modelled with the original and coarsened grids; d) loss function during optimization.

Figure 4: Solution of the coefficient inverse problem. a) True permeability, source and two measurement points (1 and 2); b) recovered permeability; c) pressure at two measurement points (data and prediction/fitting); d) loss function during 250 epochs of minimization.

Figure 5: Solution of the source inverse problem. a) True right-hand side $f$ and measurement points (black diamonds); magenta diamonds indicate the side where Dirichlet boundary condition was applied; Neumann boundary condition was applied on the other three sides; b) recovered right-hand side; c) observed data at measurement points with with noise added, $u_{obs}$ , (data) and prediction, $u_{mod}$ , (fitting); d) loss function during 100 epochs of minimization: physical loss (first term in ( 25 )) and stabilizers (later terms in ( 25 )). e) recovered right-hand side $f$ using the PINN; f) PINN loss function during 5000 epochs of minimization

PDE Solving

PDE solving develops numerical and learned methods to approximate solutions of partial differential equations across scientific applications.

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PDE Solving Differentiable Optimization PDE Inverse Problems Computational Fluid Dynamics

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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.

Abstract

PDE Solving

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