Approximate solutions of partial differential equations on unstructured grids can be represented on a graph. Treating evolutionary or static PDEs as constraints leads to large equation systems in numerical optimization. This work introduces a fully differentiable graph‑based implicit solver that embeds the constraint directly into the solution process. We evaluate it on four tasks, linear and nonlinear forward modeling for both evolutionary and static cases, inverse estimation of equation coefficients, inverse estimation of source terms, and learnable graph coarsening to speed up forward modeling. The solver’s stability and differentiability greatly reduce forward modeling complexity while efficiently handling the associated inverse problems. It is applicable to many physics and engineering domains, particularly geophysics where inverse solutions help reduce model uncertainty.
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