arXiv cs.AI· Nikolay Yavich, Alexander Ryabov, Viacheslav Naumov, Sayan Ranu, N. M. Anoop Krishnan, Evgeny Burnaev, Vladimir Vanovskiy·· 12 小时前AI 评分20
基于图的隐式求解器：前向与逆问题中的约束自动微分
Automatic Differentiation with Constraints in Forward and Inverse Problems
AI 导读
研究者提出一种全可微的图隐式求解器，将非结构网格上 PDE 近似解表示为图，并把大型方程组作为约束纳入数值优化。该求解器在四类任务上完成评估：演化与静态问题的线性/非线性前向建模、估计方程系数的逆问题、估计右端项的逆问题，以及可学习的图粗化加速前向建模。其稳定性与可微性可显著降低前向建模复杂度，并高效求解相关逆问题，适用于地球物理等需降低模型不确定性的场景。
正文 · 原文
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.
| Subjects: | Artificial Intelligence (cs.AI); Numerical Analysis (math.NA) |
| Cite as: | arXiv:2610.27888 [cs.AI] |
| (or arXiv:2610.27888v1 [cs.AI] for this version) | |
| https://doi.org/10.48550/arXiv.2610.27888 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Nikolay Yavich [view email]
[v1]
Tue, 20 Oct 2026 09:11:38 UTC (2,701 KB)
来源：arXiv cs.AI · arxiv.org