PaPi：并行积分方案让神经谱架构训练提速最高两个数量级
PaPi: Parallel Picard-Duhamel Integration for Fast Neuro-Spectral Architectures
一篇求解 PDE 的论文，把串行积分改成时间并行，训练从慢到一秒以内，还带误差证明，搞科学计算可以看看。
神经谱架构（NeuSA）是一类在频域求解含时偏微分方程的物理信息神经网络（PINN），但传统上依赖顺序 Runge-Kutta 时间步进做数值积分，限制了时间并行性。PaPi 引入基于指数积分和 Duhamel-Picard 格式的并行推理方案，并证明 Duhamel-Picard 迭代对所有流按指数收敛，给出随线性程度变化的显式误差界。实验显示训练和推理加速最高达两个数量级，低维 PDE 学习问题的训练时间缩短至一秒以内，预测精度保持有竞争力。
PaPi: Parallel Picard-Duhamel Integration for Fast Neuro-Spectral Architectures
Neuro-spectral architectures (NeuSA) are a class of physics-informed neural networks (PINNs) that have proven effective for solving time-dependent PDEs in the frequency domain. However, training and inference often rely on costly numerical integration of the learned vector field, typically using sequential Runge-Kutta time stepping, which limits temporal parallelism. We introduce PaPi, an integration scheme inspired by weakly nonlinear flows that enables time-Parallel inference through exponential integration and a Duhamel-Picard scheme. We show that the Duhamel-Picard iteration converges exponentially for all flows, and provide explicit error bounds that depend on how close the flow is to linear. Experiments show that PaPi accelerates training and inference by up to two orders of magnitude, achieving training times of under one second for low-dimensional PDE learning problems, while maintaining competitive predictive accuracy.