NEW PROGRESS IN INTELLIGENT SOLUTION OF NEURAL OPERATORS AND PHYSICS-INFORMED-BASED METHODS

Deep learning, characterized by its multi-layer neural networks, has demonstrated its capability not only to uncover hidden information in data but also to effectively address complex nonlinear problems. As a fundamental mathematical model, partial differential equations (PDEs) find wide application in describing various physical phenomena in the natural world. The amalgamation of deep learning and PDEs has given rise to the emergence of intelligent PDE solving methods based on deep learning. These methods possess several advantageous traits, including high efficiency, flexibility, and universality, which make them valuable in practical applications. This paper focuses on intelligent PDE solving methods, categorizing the solving approaches into two types based on whether they handle single or multiple problems: neural operator methods and physics-informed neural network (PINN) methods. Neural operator methods are employed to solve a class of PDE problems with the same mathematical characteristics, while PINN-based methods are used to solve single problems. The first category encompasses neural operator methods, which are utilized for solving a group of PDE problems that share similar mathematical characteristics. These methods leverage data-driven approaches and physical-constraint approaches to formulate their solutions. An in-depth analysis is conducted to examine the current research status, along with the identification of existing drawbacks in these approaches. As for PINN-based methods, this paper introduces the relevant research progress from three derivative approaches of PINN (data-optimization, model-optimization, and domain-knowledge-optimization). Finally, this paper provides a comprehensive overview of the technical roadmap for PDE intelligent solving methods. It critically evaluates the existing research limitations and proposes feasible research plans to overcome these challenges. Additionally, the paper briefly introduces the current state of intelligent solving program development and offers suggestions for future research directions. By amalgamating deep learning and PDEs, these intelligent solving methods have the potential to revolutionize various scientific and engineering domains, enabling more accurate and efficient problem-solving in complex nonlinear systems.