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

arXiv: https://arxiv.org/abs/2308.05061 · HTML: https://mlsanity.com/paper/2308.05061/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 82 references that are in the map.
- First posted on arXiv 9 Aug 2023.

## Builds on (most cited first, 30 of 40)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Visual Instruction Tuning](https://mlsanity.com/paper/2304.08485/) (2023)
- [BLIP-2: Bootstrapping Language-Image Pre-training with Frozen Image Encoders and Large Language Models](https://mlsanity.com/paper/2301.12597/) (2023)
- [Flamingo: a Visual Language Model for Few-Shot Learning](https://mlsanity.com/paper/2204.14198/) (2022)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [RT-2: Vision-Language-Action Models Transfer Web Knowledge to Robotic Control](https://mlsanity.com/paper/2307.15818/) (2023)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [MiniGPT-4: Enhancing Vision-Language Understanding with Advanced Large Language Models](https://mlsanity.com/paper/2304.10592/) (2023)
- [PaLM-E: An Embodied Multimodal Language Model](https://mlsanity.com/paper/2303.03378/) (2023)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Video-LLaMA: An Instruction-tuned Audio-Visual Language Model for Video Understanding](https://mlsanity.com/paper/2306.02858/) (2023)
- [A Survey on Multimodal Large Language Models](https://mlsanity.com/paper/2306.13549/) (2023)
- [VideoChat: Chat-Centric Video Understanding](https://mlsanity.com/paper/2305.06355/) (2023)
- [LLaMA-Adapter: Efficient Fine-tuning of Language Models with Zero-init Attention](https://mlsanity.com/paper/2303.16199/) (2023)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Multimodal Few-Shot Learning with Frozen Language Models](https://mlsanity.com/paper/2106.13884/) (2021)
- [LLaMA-Adapter V2: Parameter-Efficient Visual Instruction Model](https://mlsanity.com/paper/2304.15010/) (2023)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [PMC-VQA: Visual Instruction Tuning for Medical Visual Question Answering](https://mlsanity.com/paper/2305.10415/) (2023)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [An Empirical Study of GPT-3 for Few-Shot Knowledge-Based VQA](https://mlsanity.com/paper/2109.05014/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)

## Built on by (most cited first, 4 of 4)

- [DeepONet as a Multi-Operator Extrapolation Model: Distributed Pretraining with Physics-Informed Fine-Tuning](https://mlsanity.com/paper/2411.07239/) (2024)
- [Text-Trained LLMs Can Zero-Shot Extrapolate PDE Dynamics, Revealing a Three-Stage In-Context Learning Mechanism](https://mlsanity.com/paper/2509.06322/) (2025)
- [CompNO: A Novel Foundation Model approach for solving Partial Differential Equations](https://mlsanity.com/paper/2601.07384/) (2026)
- [Kernel Methods for Learning Operators with Multiple Inputs and Outputs](https://mlsanity.com/paper/2608.11831/) (2026)

## Abstract (arXiv)

In the growing domain of scientific machine learning, in-context operator learning has shown notable potential in building foundation models, as in this framework the model is trained to learn operators and solve differential equations using prompted data, during the inference stage without weight updates. However, the current model's overdependence on function data overlooks 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 enhanced the development of the in-context operator learning paradigm, but also created a new path for the application of language models.