# In-Context Operator Learning with Data Prompts for Differential Equation Problems Source: [In-Context Operator Learning with Data Prompts for Differential Equation Problems]() Collection index: [llms.txt]() Data updated: 2025-10-25T16:45:43.596Z Authors: Liu Yang, Siting Liu, Tingwei Meng, Stanley J. Osher Published: 2023-04-17T05:22:26.000Z ## Abstract 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Operator Learning on MFC rho-parameter family (OOD)](): Operator learning; MFC rho-parameter family (OOD); Mean Relative L2 Error: 0.004. - [Conservation-law prediction on sin-cos out-of-distribution flux function](): Conservation-law prediction; sin-cos out-of-distribution flux function; Relative L2 Error: 0.0451. - [Conservation-law prediction on tanh out-of-distribution flux function](): Conservation-law prediction; tanh out-of-distribution flux function; Relative L2 Error: 4.75. - [Conservation-law prediction on Buckley-Leverett out-of-distribution flux function](): Conservation-law prediction; Buckley-Leverett out-of-distribution flux function; Relative L2 Error: 14.18. - [Operator Learning on MFC g-parameter family (OOD)](): Operator learning; MFC g-parameter family (OOD); Mean relative L2 Error: 0.0281. - [Example-count generalization on 1D conservation law 2,000 steps k=5 examples standardized (train)](): Example-count generalization; 1D conservation law 2,000 steps k=5 examples standardized (train); MAE: 0.485. - [Example-count generalization on 1D conservation law 10,000 steps k=5 examples (standardized retraining)](): Example-count generalization; 1D conservation law 10,000 steps k=5 examples (standardized retraining); MAE: 0.48. ## Original sources and implementation [Original paper]() [PDF]()