PROSE: Predicting Multiple Operators and Symbolic Expressions using multimodal transformers.

Area of Science:

• Scientific Computing
• Machine Learning
• Differential Equations

Background:

• Neural networks approximate differential equations for scientific computing.
• Existing methods learn either the system's operator or governing equations, leading to different representations.
• Families of differential equations share characteristics, motivating a unified network representation.

Purpose of the Study:

• To develop a unified network representation for diverse differential equations.
• To enable simultaneous learning of multi-operators and governing equations.
• To enhance generalization capabilities in scientific computing tasks.

Main Methods:

• Introducing Predicting Multiple Operators and Symbolic Expressions (PROSE), a multimodal approach.
• Utilizing a novel fusion structure for simultaneous multi-operator and equation construction.
• Incorporating a language model for symbolic expression generation and equation discovery.

Main Results:

• PROSE demonstrates robust generalization across 25,600 distinct equations.
• The multimodal approach effectively handles noisy data, equation misspecification, and data imbalance.
• PROSE successfully solves differential equations, predicts future states, and generates underlying equations of motion.

Conclusions:

• PROSE offers a new operator learning framework integrating multimodal input/output and language models.
• This framework is effective for solving forward and inverse problems in differential equations.
• The approach shows significant improvements in generalization and robustness for scientific computing.