In-Context Operator Learning for Differential Equation Problems

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.