Breakeven complexity: A new perspective on neural partial differential equation solvers
For researchers and practitioners using neural PDE solvers, this work provides a more realistic evaluation framework that accounts for up-front costs and classical solver alternatives.
The paper introduces 'breakeven complexity' as a metric to evaluate neural PDE solvers by counting forward solves needed before they become cost-effective compared to error-equivalent classical solvers. Results on multiple benchmarks show neural solvers become more effective for harder problems (e.g., higher cost, dimension, Reynolds number).
Neural surrogate solvers of partial differential equations (PDEs) promise dramatic speedups over numerical methods, especially in scenarios requiring many solves. However, current accuracy-based evaluations do not fully consider two central issues: (1) neural solvers incur substantial up-front costs for data generation, training, and tuning; and (2) 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 multiple 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 become more effective as problems get harder in terms of cost, dimension, rollout, physics regime (e.g. higher Reynolds number), etc.