In the past years, the application of neural networks as an alternative to classical numerical methods to solve Partial Differential Equations has emerged as a potential paradigm shift in this century-old mathematical field. However, in terms of practical applicability, computational cost remains a substantial bottleneck. Classical approaches try to mitigate this challenge by limiting the spatial resolution on which the PDEs are defined. For neural PDE solvers, we can do better: Here, we investigate the potential of state-of-the-art quantization methods on reducing computational costs. We show that quantizing the network weights and activations can successfully lower the computational cost of inference while maintaining performance. Our results on four standard PDE datasets and three network architectures show that quantization-aware training works across settings and three orders of FLOPs magnitudes. Finally, we empirically demonstrate that Pareto-optimality of computational cost vs performance is almost always achieved only by incorporating quantization.

近年来，神经网络应用于求解偏微分方程，作为传统数值方法的替代品，这在这个有百年历史的数学领域中被认为是潜在的范式转变。然而，从实际应用的角度来看，计算成本仍然是一个重要的瓶颈。在神经PDE求解器中，我们采用了最先进的量化方法来降低计算成本的潜力。我们展示了在保持性能的同时，通过量化网络权重和激活可以成功降低推理的计算成本。我们在四个标准PDE数据集和三个网络架构上的结果表明，量化感知训练适用于各种设置和三个数量级的浮点运算。最后，我们通过实验证明，只有引入量化技术，才能几乎总是实现计算成本与性能之间的帕累托最优。

使用量化感知训练的高效神经PDE求解器