Breakeven complexity metric for neural PDE solvers

A new evaluation framework for neural PDE solvers introduces the concept of 'breakeven complexity,' a metric that counts the number of forward solves required before a learned solver becomes cost-effective compared to an error-equivalent traditional solver. The framework addresses two key issues: the substantial upfront costs of data generation, training, and tuning for neural solvers, and the ability of classical solvers to produce low-fidelity solutions at low cost. The approach uses scaling laws to allocate training budgets between data generation and model training, and discusses methods for smooth error-matching across diverse settings. This work aims to provide a more realistic assessment of neural surrogate solvers by incorporating end-to-end costs.