Classical, Machine Learning, and Hybrid Solvers for Partial Differential Equations: An Evidence-Weighted Review and Decision Framework

Abstract

Machine learning for partial differential equations raises a practical question: when are classical, learned, or hybrid solvers justified? This survey compares the three families in a decision framework organized by data availability, query volume, and certification needs. An amortization law, tested on parametric Darcy flow, shows that surrogates trained on solver-generated data break even only after many queries, beyond thirty thousand for errors below one percent. An evidence ladder, a benchmark reanalysis, and three case studies show that claims of speedup and generalization carry the weakest evidence. Solver choice should follow regime, query volume, and evidence quality, not method novelty.