Neural Fixed-Point Solvers: Learning to Iteratively Correct for Parameter-Varying Transient PDE Solving

Neural PDE solvers are typically direct predictors: they map problem data to a solution with a learned neural operator. As a result, they are agnostic to variations in PDE parameters and provide no mechanism for enforcing the governing physical constraints while solving. Classical iterative solvers address both issues by repeatedly evaluating the residual and using a preconditioner to map this feedback to a correction. Inspired by this residual-to-correction principle, we introduce the Fixed-point Neural Operator (FiNO), which uses a neural operator as a learned preconditioner. At each iteration, the PDE residual is evaluated on the current solution estimate and fed to the network, which predicts a correction that is applied repeatedly to refine the solution. This design incorporates equation feedback directly into inference and enables a shared solver to handle parameter-varying transient PDEs. Experiments across multiple problems show that the iterative solver substantially improves solution accuracy and physical consistency over direct neural operators, while providing stronger generalization across equation parameters.