[Paper Review] Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications

Zhongkai Hao, Songming Liu+5 more|arXiv (Cornell University)|Nov 15, 2022

Model Reduction and Neural Networks88 citations

TL;DR

This survey reviews how physics-informed priors are integrated with data-driven models, covering representations, methods, and applications of PIML.

ABSTRACT

Recent advances of data-driven machine learning have revolutionized fields like computer vision, reinforcement learning, and many scientific and engineering domains. In many real-world and scientific problems, systems that generate data are governed by physical laws. Recent work shows that it provides potential benefits for machine learning models by incorporating the physical prior and collected data, which makes the intersection of machine learning and physics become a prevailing paradigm. By integrating the data and mathematical physics models seamlessly, it can guide the machine learning model towards solutions that are physically plausible, improving accuracy and efficiency even in uncertain and high-dimensional contexts. In this survey, we present this learning paradigm called Physics-Informed Machine Learning (PIML) which is to build a model that leverages empirical data and available physical prior knowledge to improve performance on a set of tasks that involve a physical mechanism. We systematically review the recent development of physics-informed machine learning from three perspectives of machine learning tasks, representation of physical prior, and methods for incorporating physical prior. We also propose several important open research problems based on the current trends in the field. We argue that encoding different forms of physical prior into model architectures, optimizers, inference algorithms, and significant domain-specific applications like inverse engineering design and robotic control is far from being fully explored in the field of physics-informed machine learning. We believe that the interdisciplinary research of physics-informed machine learning will significantly propel research progress, foster the creation of more effective machine learning models, and also offer invaluable assistance in addressing long-standing problems in related disciplines.

Motivation & Objective

• Motivate the need for integrating physical laws with data-driven learning to improve robustness and generalization.
• Provide a formal framework for representing physical priors and incorporating them into ML models.
• Survey neural simulation approaches (neural solvers and neural operators) and inverse problems in PIML.
• Identify open challenges and future directions to accelerate interdisciplinary research in PIML.

Proposed method

• Define physical priors from PDEs/ODEs/SDEs, symmetry, and intuitive physics as inductive biases.
• Explain how physical priors can be embedded into data, model architectures, loss functions, optimizers, and inference.
• Present the Physics-Informed Neural Networks (PINNs) framework and its loss formulation.
• Discuss neural solvers and neural operators as neural simulation tools.
• Outline strategies for integrating priors into computer vision and reinforcement learning tasks.

Experimental results

Research questions

• RQ1 What forms of physical priors (strong to weak) are most effective for guiding ML models?
• RQ2 How can physical priors be incorporated into data, architecture, loss, optimization, and inference components of ML workflows?
• RQ3 What are the key advances, limitations, and theoretical guarantees for PINNs, DeepONet, and related methods?
• RQ4 What open problems and future directions will advance PIML in science, engineering, and AI for science?

Key findings

• Physics-informed priors can improve robustness, interpretability, and generalization by constraining learning to physically plausible solutions.
• PIML integrates data with physical laws through neural solvers like PINNs and neural operators, enabling forward and inverse problems.
• Symmetry, conservation laws, and intuitive physics provide flexible, weaker inductive biases beyond PDEs/ODEs/SDEs.
• A broad spectrum of applications exists across scientific domains, computer vision, and reinforcement learning, highlighting interdisciplinary potential.