Graph based differentiable solver improves modeling and inverse problem solving
Automatic Differentiation with Constraints in Forward and Inverse Problems
Artificial Intelligence
Summary
Many scientific and engineering problems can be described using equations that are hard to solve, especially when you want to figure out the unknown parts by working backward from data. The authors created a new computer method that represents these problems on graphs and can automatically compute how changes affect the outcome. This makes it easier and faster to solve both forward problems (predicting what will happen) and inverse problems (finding unknown parameters) while keeping the calculations stable. Their approach can be especially helpful in fields like geophysics, where understanding underground structures depends on solving these tricky problems.
What this means in practice
- •For geophysics engineers: Reduce uncertainty in underground models by efficiently solving inverse problems to estimate physical parameters from measurements.
- •For computational physics developers: Speed up simulations of complex physical systems by using learnable graph coarsening embedded in differentiable PDE solvers.
Authors
Nikolay Yavich, Alexander Ryabov, Viacheslav Naumov, Sayan Ranu, N. M. Anoop Krishnan, Evgeny Burnaev, Vladimir Vanovskiy
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.