Foundation models, such as large language models, have demonstrated success
in addressing various language and image processing tasks. In this work, we
introduce a multi-modal foundation model for scientific problems, named
PROSE-PDE. Our model, designed for bi-modality to bi-modality learning, is a
multi-operator learning approach which can predict future states of
spatiotemporal systems while concurrently learning the underlying governing
equations of the physical system. Specifically, we focus on multi-operator
learning by training distinct one-dimensional time-dependent nonlinear constant
coefficient partial differential equations, with potential applications to many
physical applications including physics, geology, and biology. More
importantly, we provide three extrapolation studies to demonstrate that
PROSE-PDE can generalize physical features through the robust training of
multiple operators and that the proposed model can extrapolate to predict PDE
solutions whose models or data were unseen during the training. Furthermore, we
show through systematic numerical experiments that the utilization of the
symbolic modality in our model effectively resolves the well-posedness problems
with training multiple operators and thus enhances our model's predictive
capabilities.

我们介绍了一种用于科学问题的多模态基础模型 PROSE-PDE，它是一种多运算符学习方法，可以预测时空系统的未来状态，并同时学习物理系统的底层控制方程。我们通过三个外推研究证明了 PROSE-PDE 可以通过训练多个运算符来泛化物理特征，且该模型能够推断出在训练过程中未见过模型或数据的偏微分方程解。此外，通过系统数值实验，我们展示了我们模型中符号模态的有效利用解决了训练多个运算符的完整性问题，从而增强了我们模型的预测能力。