When Newton Helps Newton: Learning Solver-Guided Warm Starts for PDE Solves

Nonlinear partial differential equations (PDEs) solved with the finite element method (FEM) require multiple Newton-Raphson iterations. The computational cost scales with iteration count, especially for large systems. Using learned warm-starts is a promising approach to reduce the iteration count and computational cost. Existing approaches for learning warm-starts rely on physics-informed losses or precomputed solution data. We propose a solver-guided training approach that leverages the fixed-point consistency of the Newton-Raphson method to learn warm-starts for multiple instances of a PDE class. Specifically, we show that a limited solver iteration budget in this training strategy results in a higher warm-start benefit despite a lower point-wise accuracy against the PDE solutions. The warm-starts learned with this approach produce converged solutions with fewer Newton iterations translating to computational time speedup. We present the results on a moderately nonlinear elliptic PDE with extensions to strongly nonlinear problems as future work.