Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications

TL;DR

Physics‑informed machine learning (PIML) integrates empirical data with physical priors such as $\mathcal{F}(u;\theta)(\boldsymbol{x})=0$, symmetry, and intuitive physics to create physically plausible models. The paper surveys problem formulations, representations of physical priors, and a spectrum of methods, including PINNs, neural operators (e.g., DeepONet, Green's function learning, FNO), and domain‑decomposition approaches, along with theory and applications across fluids, materials, inverse design, computer vision/graphics, and RL. Key contributions include a unified view of incorporating priors into data, architecture, losses, optimizers, and inference; a distillation of neural solvers, operators, and theory; and a roadmap of open challenges in optimization, high‑dimensional PDEs, data efficiency, and real‑world deployment. The work underscores the potential of physics‑biased AI to improve robustness, generalization, and interpretability while catalyzing interdisciplinary advances in science and engineering.

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.

Figures (3)

• Figure 1: An overview of physics-informed machine learning. We review various methods of incorporating physical prior knowledge into machine learning models, ranging from strong to weak forms, such as PDEs/ODEs/SDEs, symmetry, and intuitive physics. These physical priors can be incorporated into different aspects of machine learning models, such as data, model architecture, loss function, optimizer, and inference algorithm. We also highlight different applications of physics-informed machine learning in tasks such as neural simulation, inverse problems, CV/NLP, and RL/control. Finally, we identify some significant areas for exploration in the PIML field, such as physics-informed optimizers and physics-informed inference methods.
• Figure 2: A chronological overview of important methods for neural simulation (neural solver and neural operator) and inverse problems (inverse design) of physics-informed machine learning. The earliest work could be traced back to dissanayake1994neural.
• Figure 3: The Architecture of FNOs.

Theorems & Definitions (7)

• Theorem 1: Universal Approximation Theorem for DeepONet chen1995universallu2021learning
• Theorem 2: yu2021arbitrary
• Theorem 3: yu2021arbitrary
• Theorem 4: shin2020convergence
• Theorem 5: shin2020convergence
• Theorem 6: lanthaler2022error
• Theorem 7: lanthaler2022error