Automatic Differentiation with Constraints in Forward and Inverse Problems

Key points

• Numerical optimization involving stiff evolutionary and static PDEs often requires solving a large system of equations, which can be treated as a constraint.1
• In this work, we present a fully differentiable graph-based implicit solver that incorporates this constraint.1
• 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.1
• The stability and differentiability of our solver enable significant reductions in the complexity of forward modeling while efficiently addressing related inverse problems.1

Sources (1)

• [1]Automatic Differentiation with Constraints in Forward and Inverse Problems
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 20, 09:11 AM

  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.

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