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 learning framework to address these issues. A new spatiotemporal adaptation is proposed to generalize any Fourier Neural Operator (FNO) variant to learn maps between Bochner spaces, which can perform an arbitrary-length temporal super-resolution for the first time. To better exploit this capacity, a new paradigm is proposed to refine the commonly adopted end-to-end neural operator training and evaluations with the help from the wisdom from traditional numerical PDE theory and techniques. Specifically, in the learning problems for the turbulent flow modeled by the Navier-Stokes Equations (NSE), the proposed paradigm trains an FNO only for a few epochs. Then, only the newly proposed spatiotemporal spectral convolution layer is fine-tuned without the frequency truncation. The spectral fine-tuning loss function uses a negative Sobolev norm for the first time in operator learning, defined through a reliable functional-type a posteriori error estimator whose evaluation is exact thanks to the Parseval identity. Moreover, unlike the difficult nonconvex optimization problems in the end-to-end training, this fine-tuning loss is convex. Numerical experiments on commonly used NSE benchmarks demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers under certain conditions. The source code is publicly available at https://github.com/scaomath/torch-cfd.

Transmission lines’ non-negligible distributed capacitance can increase the risk of broadband resonance in power systems. However, the distributed parameter characteristics and frequency-dependent characteristics are commonly ignored in the transmission line modeling during the analysis of resonance, which may lead to inaccurate stability assessment and pseudo oscillations in simulations. In this article, a harmonic frequency range transmission line model suitable for real-time simulation (RTS) is proposed. First, a simplified calculation method of parameters based on system structures is expanded. Afterward, a detailed equivalent circuit of the multi-phase model is implemented in RTS to meet the requirements of decoupling and parallel computing. Finally, the electrical parameters are modified based on the harmonic frequency to improve the Bergeron model and the proposed model (PM) is developed accordingly. Simulation results show that the PM is highly suitable for time-domain simulation and high-order harmonic analysis.

In this paper, we apply the self-attention from the state-of-the-art Transformer in Attention Is All You Need for the first time to a data-driven operator learning problem related to partial differential equations. An effort is put together to explain the heuristics of, and to improve the efficacy of the attention mechanism. By employing the operator approximation theory in Hilbert spaces, it is demonstrated for the first time that the softmax normalization in the scaled dot-product attention is sufficient but not necessary. Without softmax, the approximation capacity of a linearized Transformer variant can be proved to be comparable to a Petrov-Galerkin projection layer-wise, and the estimate is independent with respect to the sequence length. A new layer normalization scheme mimicking the Petrov-Galerkin projection is proposed to allow a scaling to propagate through attention layers, which helps the model achieve remarkable accuracy in operator learning tasks with unnormalized data. Finally, we present three operator learning experiments, including the viscid Burgers' equation, an interface Darcy flow, and an inverse interface coefficient identification problem. The newly proposed simple attention-based operator learner, Galerkin Transformer, shows significant improvements in both training cost and evaluation accuracy over its softmax-normalized counterparts.

With the maturity of modern science and technology, such as networks and computers, Internet of Things has been widely used in various fields of industry, opening up a new situation for the development of the industry and creating a broader development platform. This paper systematically analyzes the characteristics and changes of the commodity structure, regional structure, market structure, and main structure of export management of agricultural products. The proposed algorithm uses computer technology, network technology, and remote communication technology, electronic, digital, and network—the entire business process of business. The empirical analysis shows that the structure of agricultural product export trade reflects the endowment and comparative advantage of agricultural resources in China. The proposed blockchain technology supports and guides agricultural export enterprises to develop their own brands, strengthen the quality and safety management of agricultural products, and innovate and expand the policy support system for optimizing the structure of agricultural product export trade. By optimizing the export mode of agricultural trade, we can achieve the purpose of increasing the export volume of agricultural products and the total value of trade exports. The experimental results show the effectiveness of the proposed algorithm, which can greatly improve the volume of trade exports and the total value of trade exports.