成本-精度权衡：神经算子与经典数值求解器
Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver
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中文总结 AI 辅助
本研究通过可复现基准，比较神经算子与经典数值求解器在给定精度下的成本，发现神经算子在低至中等精度时具优势，但高精度下经典求解器更优，且经典求解器对验证与高精度计算仍不可或缺。
中文摘要 AI 辅助
神经算子是数据驱动的模型，学习从参数化偏微分方程的输入（如空间变化系数、初始条件、强迫项、边界条件或几何形状）到解场或感兴趣量的映射。一旦训练完成，它们可以在需要对不同输入进行重复评估的多查询设置中作为经典数值求解器的替代模型。我们探讨了在给定精度下，神经算子替代模型何时以及为何在成本方面优于经典数值求解器的问题。我们关注训练后的多查询极限，其中数据获取和训练成本被视为固定且完全摊销的。即使在这种对神经算子有利的刻意设定下，也存在经典求解器优于替代模型的场景。我们通过一项可复现的基准研究，比较了神经算子替代模型与经典数值求解器的成本-精度性能，该研究在计算科学与工程的代表性问题上，将神经算子与问题匹配的经典求解器进行了对比，重点关注预测误差、每次查询的浮点成本和墙钟运行时间。神经算子在低至中等精度要求下最具竞争力。其浮点成本优势强烈依赖于问题结构，当它们避免时间迭代或非线性迭代，或预测缩减的感兴趣量而非完整解场时，这种优势就会出现。额外的墙钟加速源于适合现代硬件的密集张量运算。随着目标精度的提高，使用神经算子达到所需精度变得越来越具有挑战性，而经典求解器在此场景下优于替代模型；因此，经典求解器在验证和高精度计算方面仍将保持重要地位。
英文摘要
Neural operators are data-driven models that learn mappings from inputs that parameterize partial differential equations, such as spatially varying coefficients, initial conditions, forcing terms, boundary conditions, or geometries, to solution fields or quantities of interest. Once trained, they can serve as surrogates for classical numerical solvers in many-query settings that require repeated evaluations for varying inputs. We address the question of when, and then why, neural operator surrogates outperform classical numerical solvers, in terms of cost for a given accuracy. We focus on the post-training, many-query limit, in which data-acquisition and training costs are treated as fixed and fully amortized. Even in this deliberately favorable regime for neural operators, there are regimes in which classical solvers outperform the surrogate models. We compare the cost-accuracy performance of neural operator surrogates and classical numerical solvers through a reproducible benchmark study comparing neural operators with problem-matched classical solvers on representative problems in computational science and engineering, focusing on prediction error, per-query floating-point cost, and wall-clock runtime. Neural operators are most competitive at low-to-moderate accuracy requirements. Their floating-point cost advantage depends strongly on the problem structure, arising when they avoid temporal or nonlinear iterations or predict a reduced quantity of interest rather than a full solution field. Additional wall-clock speedups result from dense tensor operations that are well suited to modern hardware. As the target accuracy is tightened, achieving the required accuracy with neural operators becomes increasingly challenging, and classical solvers outperform surrogates in this regime; thus classical solvers will remain important for verification and high-accuracy computation.
发表机构
- Center for Machine Learning Research, Peking University（北京大学机器学习研究中心）
- California Institute of Technology（加州理工学院）
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