This paper introduces a new neural-network-based approach, namely IN-context Differential Equation Encoder-Decoder (INDEED), to simultaneously learn operators from data and apply it to new questions during the inference stage, without any weight update. Existing methods are limited to using a neural network to approximate a specific equation solution or a specific operator, requiring retraining when switching to a new problem with different equations. By training a single neural network as an operator learner, we can not only get rid of retraining (even fine-tuning) the neural network for new problems, but also leverage the commonalities shared across operators so that only a few demos are needed when learning a new operator. Our numerical results show the neural network's capability as a few-shot operator learner for a diversified type of differential equation problems, including forward and inverse problems of ODEs and PDEs, and also show that it can generalize its learning capability to operators beyond the training distribution, even to an unseen type of operator.

翻译：本文提出了一种基于神经网络的新方法，即上下文微分方程编码器-解码器（INDEED），可在无需任何权重更新的情况下，同时从数据中学习算子并应用于推理阶段的新问题。现有方法局限于使用神经网络近似特定方程的解或特定算子，当切换至不同方程的新问题时需要重新训练。通过训练单一神经网络作为算子学习器，我们不仅能避免针对新问题重新训练（甚至微调）该网络，还能利用各算子间的共性，从而在学习新算子时仅需少量示例。数值结果表明，该网络作为少样本算子学习器，能够处理多种类型的微分方程问题（包括常微分方程和偏微分方程的正问题与反问题），且其学习能力可泛化至训练分布之外的算子，甚至包括未曾见过的算子类型。