Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new Spatiotemporal Fourier Neural Operator (SFNO) that learns maps between Bochner spaces, and a new learning framework to address these issues. This new paradigm leverages wisdom from traditional numerical PDE theory and techniques to refine the pipeline of commonly adopted end-to-end neural operator training and evaluations. Specifically, in the learning problems for the turbulent flow modeling by the Navier-Stokes Equations (NSE), the proposed architecture initiates the training with a few epochs for SFNO, concluding with the freezing of most model parameters. Then, the last linear spectral convolution layer is fine-tuned without the frequency truncation. The optimization uses a negative Sobolev norm for the first time as the loss in operator learning, defined through a reliable functional-type \emph{a posteriori} error estimator whose evaluation is almost exact thanks to the Parseval identity. This design allows the neural operators to effectively tackle low-frequency errors while the relief of the de-aliasing filter addresses high-frequency errors. Numerical experiments on commonly used benchmarks for the 2D NSE demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers.

翻译：近年来，算子型神经网络在逼近时空偏微分方程解方面展现出有前景的结果。然而，这类神经网络通常需要高昂的训练成本，且往往难以达到许多科学与工程领域所要求的精度标准。本文提出一种在Bochner空间之间学习映射关系的新型时空傅里叶神经算子，并构建了新的学习框架以解决上述问题。该新范式借鉴传统数值偏微分方程理论与技术的思想，对当前普遍采用的端到端神经算子训练与评估流程进行精细化改进。具体而言，在基于Navier-Stokes方程的湍流建模学习任务中，所提架构首先对SFNO进行少量周期训练，随后冻结大部分模型参数，最后对未施加频率截断的线性谱卷积层进行微调。优化过程首次采用负Sobolev范数作为算子学习的损失函数，该范数通过可靠泛函型后验误差估计器定义，并借助Parseval恒等式实现近乎精确的评估。此设计使神经算子能有效处理低频误差，而解混叠滤波的解除则有助于应对高频误差。在二维NSE常用基准测试上的数值实验表明，相较于端到端评估方法与传统数值偏微分方程求解器，本方法在计算效率与精度方面均取得显著提升。