Automatic Differentiation with Constraints in Forward and Inverse Problems
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.
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BibTeX
@misc{yavich2026automatic,
title = {{Automatic Differentiation with Constraints in Forward and Inverse Problems}},
author = {Nikolay Yavich and Alexander Ryabov and Viacheslav Naumov and Sayan Ranu and N. M. Anoop Krishnan and Evgeny Burnaev and Vladimir Vanovskiy},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27888},
eprint = {2610.27888},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27888},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Automatic Differentiation with Constraints in Forward and Inverse Problems
AU - Nikolay Yavich
AU - Alexander Ryabov
AU - Viacheslav Naumov
AU - Sayan Ranu
AU - N. M. Anoop Krishnan
AU - Evgeny Burnaev
AU - Vladimir Vanovskiy
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.27888
UR - https://arxiv.org/abs/2610.27888
AB - 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.
ER -