Fine-Tune Language Models as Multi-Modal Differential Equation Solvers

In the growing domain of scientific machine learning, in-context operator learning has shown notable potential in learning operators and solving differential equations using prompted data, during the inference stage without weight updates. However, the current model's overdependence on function data, may inadvertently overlook the invaluable human insight into the operator. To address this, we present a transformation of in-context operator learning into a multi-modal paradigm. In particular, we take inspiration from the recent success of large language models, and propose using "captions" to integrate human knowledge about the operator, expressed through natural language descriptions and equations. Also, we introduce a novel approach to train a language-model-like architecture, or directly fine-tune existing language models, for in-context operator learning. We beat the baseline on single-modal learning tasks, and also demonstrated the effectiveness of multi-modal learning in enhancing performance and reducing function data requirements. The proposed method not only significantly improves in-context operator learning, but also creates a new path for the application of language models.