Breakeven complexity: A new perspective on neural partial differential equation solvers

Neural surrogate solvers of partial differential equations (PDEs) promise dramatic speedups over numerical methods, especially in scenarios requires many solves. However, current accuracy-based evaluations do not fully consider two central issues: (i) neural solvers incur substantial up-front costs for data generation, training, and tuning; and (ii) classical solvers can also generate low-fidelity solutions at a sufficiently low simulation cost. To explicitly account for these realities and fully incorporate end-to-end costs, we propose an evaluation framework centered on breakeven complexity, a metric that counts the forward solves before a learned solver is cost-effective relative to an error-equivalent traditional solver. To evaluate this measure, we apply scaling laws to determine how much training budget to allocate to data generation and discuss how to achieve smooth error-matching in diverse settings. We evaluate the breakeven complexity of many modern neural PDE solvers on three PDEs on 2D periodic domains from APEBench and a novel benchmark of flows past multiple obstacles generated by the GPU-native PyFR code. Among other findings, our results suggest that neural PDE solvers will be more effective on problems as problems get harder in terms of cost, dimension, rollout, physics regime (e.g. higher Re), etc.