# MaD-Scientist: AI-based Scientist solving Convection-Diffusion-Reaction Equations Using Massive PINN-Based Prior Data

arXiv: https://arxiv.org/abs/2410.06442 · HTML: https://mlsanity.com/paper/2410.06442/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 46 references that are in the map.
- First posted on arXiv 9 Oct 2024.

## Builds on (most cited first, 26 of 26)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [What Can Transformers Learn In-Context? A Case Study of Simple Function Classes](https://mlsanity.com/paper/2208.01066/) (2022)
- [TabPFN: A Transformer That Solves Small Tabular Classification Problems in a Second](https://mlsanity.com/paper/2207.01848/) (2022)
- [Large Language Models Are Zero-Shot Time Series Forecasters](https://mlsanity.com/paper/2310.07820/) (2023)
- [Transformers Can Do Bayesian Inference](https://mlsanity.com/paper/2112.10510/) (2021)
- [Trained Transformers Learn Linear Models In-Context](https://mlsanity.com/paper/2306.09927/) (2023)
- [Transformers as Statisticians: Provable In-Context Learning with In-Context Algorithm Selection](https://mlsanity.com/paper/2306.04637/) (2023)
- [Mathematical Capabilities of ChatGPT](https://mlsanity.com/paper/2301.13867/) (2023)
- [Why Can GPT Learn In-Context? Language Models Implicitly Perform Gradient Descent as Meta-Optimizers](https://mlsanity.com/paper/2212.10559/) (2022)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior](https://mlsanity.com/paper/2306.00258/) (2023)
- [UniAudio: An Audio Foundation Model Toward Universal Audio Generation](https://mlsanity.com/paper/2310.00704/) (2023)
- [In-Context Operator Learning with Data Prompts for Differential Equation Problems](https://mlsanity.com/paper/2304.07993/) (2023)
- [In-Context Learning through the Bayesian Prism](https://mlsanity.com/paper/2306.04891/) (2023)
- [Statistical Foundations of Prior-Data Fitted Networks](https://mlsanity.com/paper/2305.11097/) (2023)
- [Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks](https://mlsanity.com/paper/2310.09528/) (2023)
- [Parameterized Physics-informed Neural Networks for Parameterized PDEs](https://mlsanity.com/paper/2408.09446/) (2024)
- [PDE Generalization of In-Context Operator Networks: A Study on 1D Scalar Nonlinear Conservation Laws](https://mlsanity.com/paper/2401.07364/) (2024)
- [Towards Few-Shot Adaptation of Foundation Models via Multitask Finetuning](https://mlsanity.com/paper/2402.15017/) (2024)
- [A Perspective on Machine Learning Methods in Turbulence Modelling](https://mlsanity.com/paper/2010.12226/) (2020)
- [Spectral operator learning for parametric PDEs without data reliance](https://mlsanity.com/paper/2310.02013/) (2023)

## Abstract (arXiv)

Large language models (LLMs), like ChatGPT, have shown that even trained with noisy prior data, they can generalize effectively to new tasks through in-context learning (ICL) and pre-training techniques. Motivated by this, we explore whether a similar approach can be applied to scientific foundation models (SFMs). Our methodology is structured as follows: (i) we collect low-cost physics-informed neural network (PINN)-based approximated prior data in the form of solutions to partial differential equations (PDEs) constructed through an arbitrary linear combination of mathematical dictionaries; (ii) we utilize Transformer architectures with self and cross-attention mechanisms to predict PDE solutions without knowledge of the governing equations in a zero-shot setting; (iii) we provide experimental evidence on the one-dimensional convection-diffusion-reaction equation, which demonstrate that pre-training remains robust even with approximated prior data, with only marginal impacts on test accuracy. Notably, this finding opens the path to pre-training SFMs with realistic, low-cost data instead of (or in conjunction with) numerical high-cost data. These results support the conjecture that SFMs can improve in a manner similar to LLMs, where fully cleaning the vast set of sentences crawled from the Internet is nearly impossible.