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

2026-05-14

AI Summary

This work addresses the common oversight in evaluating neural PDE solvers—namely, the neglect of training costs and the absence of fair cost-effectiveness comparisons against traditional low-accuracy numerical methods. To remedy this, we introduce a novel evaluation framework centered on “break-even complexity,” which holistically accounts for data generation, training overhead, and the computational cost of conventional solvers achieving equivalent error levels. Guided by scaling laws, our framework also informs optimal training budget allocation. Using neural surrogates, GPU-native PyFR-generated multi-obstacle flow benchmarks, and two-dimensional periodic-domain PDE problems from APEBench, we quantify break-even points across diverse scenarios. Our analysis reveals that neural solvers exhibit practical deployment advantages particularly in high-dimensional, high-Reynolds-number, or otherwise computationally expensive regimes.

Abstract

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.

Problem

• neural PDE solvers
• breakeven complexity
• evaluation framework
• computational cost
• accuracy-cost trade-off

Innovation

• breakeven complexity
• neural PDE solvers
• scaling laws
• error-matching
• cost-effective evaluation