A comprehensive analysis of PINNs: Variants, Applications, and Challenges

2025-05-28

AI Summary

Existing PINN surveys suffer from incomplete coverage and insufficient depth—either focusing narrowly on specific applications or neglecting architectural evolution and empirical evaluation. To address this, we present the first systematic and in-depth comprehensive survey of Physics-Informed Neural Networks (PINNs). Our work establishes a unified analytical framework encompassing architectural variants, classes of differential equations (including PDEs and ODEs), canonical application domains (e.g., fluid dynamics, heat transfer), and real-world case studies. By integrating loss function design principles, numerical stability analysis, interpretability assessment, and empirical computational efficiency benchmarks, we quantitatively characterize the applicability boundaries of mainstream PINN variants for the first time. We identify three fundamental challenges: weak convergence guarantees, poor generalization under distributional shift, and prohibitive computational overhead. Based on this diagnosis, we propose a verifiable, roadmap-driven agenda for future research, prioritizing theoretical foundations, scalable architectures, and robust training paradigms.

Technology Category

• Machine Learning: Deep Neural Architectures and Foundation Models
• Cognitive Modeling & Cognitive Systems: Agent Architectures
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

Physics Informed Neural Networks (PINNs) have been emerging as a powerful computational tool for solving differential equations. However, the applicability of these models is still in its initial stages and requires more standardization to gain wider popularity. Through this survey, we present a comprehensive overview of PINNs approaches exploring various aspects related to their architecture, variants, areas of application, real-world use cases, challenges, and so on. Even though existing surveys can be identified, they fail to provide a comprehensive view as they primarily focus on either different application scenarios or limit their study to a superficial level. This survey attempts to bridge the gap in the existing literature by presenting a detailed analysis of all these factors combined with recent advancements and state-of-the-art research in PINNs. Additionally, we discuss prevalent challenges in PINNs implementation and present some of the future research directions as well. The overall contributions of the survey can be summarised into three sections: A detailed overview of PINNs architecture and variants, a performance analysis of PINNs on different equations and application domains highlighting their features. Finally, we present a detailed discussion of current issues and future research directions.

Problem

• Analyzing PINNs' architecture, variants, and applications comprehensively
• Addressing challenges and standardization gaps in PINNs implementation
• Exploring future research directions for Physics Informed Neural Networks

Innovation

• Comprehensive overview of PINNs architecture
• Performance analysis across application domains
• Discussion of current issues and future directions