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

arXiv: https://arxiv.org/abs/2605.15399 · HTML: https://mlsanity.com/paper/2605.15399/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 34 references that are in the map.
- First posted on arXiv 14 May 2026.

## Builds on (most cited first, 22 of 22)

- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [APEBench: A Benchmark for Autoregressive Neural Emulators of PDEs](https://mlsanity.com/paper/2411.00180/) (2024)
- [Parameters vs FLOPs: Scaling Laws for Optimal Sparsity for Mixture-of-Experts Language Models](https://mlsanity.com/paper/2501.12370/) (2025)
- [Machine Learning For Elliptic PDEs: Fast Rate Generalization Bound, Neural Scaling Law and Minimax Optimality](https://mlsanity.com/paper/2110.06897/) (2021)
- [Walrus: A Cross-Domain Foundation Model for Continuum Dynamics](https://mlsanity.com/paper/2511.15684/) (2025)
- [DISCO: learning to DISCover an evolution Operator for multi-physics-agnostic prediction](https://mlsanity.com/paper/2504.19496/) (2025)
- [FlowBench: A Large Scale Benchmark for Flow Simulation over Complex Geometries](https://mlsanity.com/paper/2409.18032/) (2024)
- [EddyFormer: Accelerated Neural Simulations of Three-Dimensional Turbulence at Scale](https://mlsanity.com/paper/2510.24173/) (2025)
- [A Benchmarking Framework for AI models in Automotive Aerodynamics](https://mlsanity.com/paper/2507.10747/) (2025)
- [Neural Networks to solve Partial Differential Equations: a Comparison with Finite Elements](https://mlsanity.com/paper/2201.03269/) (2022)
- [Pre-Generating Multi-Difficulty PDE Data for Few-Shot Neural PDE Solvers](https://mlsanity.com/paper/2512.00564/) (2025)

## Abstract (arXiv)

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.