Discretized Governing Equation

Discretized governing equations represent a crucial area of research bridging numerical methods and machine learning for solving partial differential equations (PDEs). Current efforts focus on integrating neural networks, particularly convolutional and feed-forward architectures, within established numerical schemes like finite element methods to improve efficiency and accuracy, often incorporating physics-informed learning techniques to constrain solutions and enhance physical realism. This approach aims to overcome limitations of traditional numerical methods, such as computational cost and difficulty handling complex geometries or discontinuous data, leading to more efficient and accurate simulations across diverse scientific and engineering domains.