In-context operator networks (ICON) are a class of operator learning methods based on the novel architectures of foundation models. Trained on a diverse set of datasets of initial and boundary conditions paired with corresponding solutions to ordinary and partial differential equations (ODEs and PDEs), ICON learns to map example condition-solution pairs of a given differential equation to an approximation of its solution operator. Here, we present a probabilistic framework that reveals ICON as implicitly performing Bayesian inference, where it computes the mean of the posterior predictive distribution over solution operators conditioned on the provided context, i.e., example condition-solution pairs. The formalism of random differential equations provides the probabilistic framework for describing the tasks ICON accomplishes while also providing a basis for understanding other multi-operator learning methods. This probabilistic perspective provides a basis for extending ICON to \emph{generative} settings, where one can sample from the posterior predictive distribution of solution operators. The generative formulation of ICON (GenICON) captures the underlying uncertainty in the solution operator, which enables principled uncertainty quantification in the solution predictions in operator learning.

SINCE WE ARE FIRMLY CONVINCED that ideas like people have histories and can only be fully understood in the context of those histories, we will begin by discussing the underlying ideas that motivated us to initiate the Steamer effort. Without richer and more detailed understandings of the nature of these models, instructional applications will be severely limited. Graphical Interfaces for Interactave Inspectable Simulatzons - We believe that graphical interfaces to simulations of physical systems deserve extensive exploration. They make possible new types of instructional interactions by allowing one to control, manipulate, and monitor simulations of dynamic systems at many different hierarchical levels The key idea in Steamer is the conception of an znteractive inspectable simulation. We have consistently sought to make the system inspectable.

In this article, we investigate the possibility of cross-language communication using a synergy of words and pictures on mobile devices. On the one hand, communicating with only pictures is in itself a very powerful strategy, but is limited in expressiveness. On the other hand, words can express everything you could wish to say, but they are cumbersome to work with on mobile devices and need to be translated in order for their meaning to be understood. Automatic translations can contain errors that pervert the communication process, and this may undermine the users' confidence when expressing themselves across language barriers. Our idea is to create a user interface for cross-language communication that uses pictures as the primary mode of input, and words to express the detailed meaning.