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

NeurIPS 2026 · Probabilistic and scientific modeling › Differential equations and PDEs · approach: Neural PDE solvers · 0 citations (OpenAlex)

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Proposes breakeven complexity, a cost-aware metric counting forward solves before a neural PDE surrogate (including its data generation and training cost) beats an error-matched low-fidelity classical solver, and evaluates eight neural solvers on APEBench periodic PDEs and a new PyFR-based flow-past-obstacles benchmark, BreakFlow.

Problem

Accuracy-only evaluations of neural PDE surrogates ignore up-front data generation and training costs and do not compare against low-fidelity classical solvers that are cheap enough to match the surrogate's error, so it is unclear when learned solvers are worth their cost.

Approach

Define breakeven complexity N* = B / max(C_delta_B - C_inf, 0), where B is total up-front cost (data generation plus training) and delta_B is the cheapest classical solver configuration matching the surrogate's error; make it practical via scaling-law allocation of training budget between data and optimization and by coarsening resolution and timestep (CFL-constrained) to error-match, with worst-c

Key results

• Breakeven complexity for APEBench periodic tasks often reaches hundreds of thousands of forward solves (e.g., FFNO on Navier-Stokes at 8K budget about 203,692 average-case), while BreakFlow requires only a few thousand.
• Poseidon-L is the worst Poseidon model on most tasks under equalized training cost, and FFNO is the most cost-effective at the highest budget on BreakFlow despite Poseidon/DPOT dominating in test error.
• Breakeven complexity decreases as problems get harder (higher dimension for Kuramoto-Sivashinsky, longer rollouts for Gray-Scott, higher Reynolds number for BreakFlow), while test error increases.

Surprising findings

• Models with the best test error on BreakFlow (Poseidon and DPOT) are not much better than HalfWalrus and FFNO in breakeven complexity, with FFNO most cost-effective at the highest budget.
• Neural solvers become more cost-effective as problems get harder, the opposite of what nRMSE alone suggests.

Limitations

• Breakeven complexity depends on the chosen classical baseline, implementation, and hardware, so it is a deployment-specific calibration rather than a universal ranking.
• Assumes simulation and inference cost is roughly constant across parameters theta, which may fail when theta affects problem difficulty.
• Only wall-clock cost on a single GPU type is used; heterogeneous hardware and CPU-based classical solvers are not studied.

Experimental setup

Datasets: APEBench Navier-Stokes APEBench Kuramoto-Sivashinsky APEBench Gray-Scott BreakFlow

Benchmarks: APEBench Navier-Stokes APEBench Kuramoto-Sivashinsky APEBench Gray-Scott BreakFlow

Metrics: normalized RMSE (average and worst-case) breakeven complexity (average and worst-case) wall-clock time

Baselines: FFNO EddyFormer DPOT DISCO Poseidon-T Poseidon-B Poseidon-L HalfWalrus

Builds on: neural operators Chinchilla scaling laws APEBench Exponax

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