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This paper introduces PDEformer-1, a versatile neural solver capable of simultaneously addressing various partial differential equations (PDEs). With the PDE represented as a computational graph, we facilitate the seamless integration of symbolic and numeric information inherent in a PDE. A graph Transformer and an implicit neural representation (INR) are employed subsequently to generate mesh-free predicted solutions. We generated a dataset with up to three million samples involving diverse one-dimensional PDEs to pretrain our model. Compared with baseline models trained specifically on benchmark datasets, our pretrained model achieves comparable accuracy via zero-shot inference, and the advantage expands after finetuning. For PDEs new or unseen in the pretraining stage, our model can adapt quickly by finetuning on a relatively small set of examples from the target equation. Additionally, PDEformer-1 demonstrates promising results in the inverse problem of PDE scalar coefficient recovery and coefficient field recovery.

Customer services are critical to all companies, as they may directly connect to the brand reputation. Due to a great number of customers, e-commerce companies often employ multiple communication channels to answer customers' questions, for example, chatbot and hotline. On one hand, each channel has limited capacity to respond to customers' requests, on the other hand, customers have different preferences over these channels. The current production systems are mainly built based on business rules, which merely considers tradeoffs between resources and customers' satisfaction. To achieve the optimal tradeoff between resources and customers' satisfaction, we propose a new framework based on deep reinforcement learning, which directly takes both resources and user model into account. In addition to the framework, we also propose a new deep-reinforcement-learning based routing method-double dueling deep Q-learning with prioritized experience replay (PER-DoDDQN). We evaluate our proposed framework and method using both synthetic and a real customer service log data from a large financial technology company. We show that our proposed deep-reinforcement-learning based framework is superior to the existing production system. Moreover, we also show our proposed PER-DoDDQN is better than all other deep Q-learning variants in practice, which provides a more optimal routing plan. These observations suggest that our proposed method can seek the trade-off where both channel resources and customers' satisfaction are optimal., Comment: 13 pages, 7 figures