This paper introduces a new neural-network-based approach, namely In-Context Operator Networks (ICON), to simultaneously learn operators from the prompted 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 in the prompt 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 ordinary differential equations (ODEs), partial differential equations (PDEs), and mean-field control (MFC) problems, and also show that it can generalize its learning capability to operators beyond the training distribution.

翻译：本文提出一种基于神经网络的新方法——上下文算子网络（ICON），该方法可在推理阶段同时从提示数据中学习算子并应用于新问题，无需任何权重更新。现有方法局限于使用神经网络近似特定方程解或特定算子，当切换至不同方程的新问题时需重新训练。通过训练单一神经网络作为算子学习器，我们不仅可免除针对新问题重新训练（甚至微调）神经网络的步骤，还能利用不同算子间的共性，使得学习新算子时仅需提示中的少量示范样本。数值实验表明，该神经网络作为少样本算子学习器，能处理多种类型的微分方程问题，包括常微分方程（ODE）的正向/逆向问题、偏微分方程（PDE）问题以及平均场控制（MFC）问题，同时展现出跨训练分布算子的泛化学习能力。