Purpose
The primary objective of this particular research is to mainly explore the effectiveness of machine learning approaches in the process of solving the nonlinear partial differential equations (PDEs). The paper will also offer a comparative evaluation of some of the machine learning tools, including neural networks, support vector machines, and deep learning models, to evaluate how these tools have improved in the aspects of accuracy, efficiency, and scalability to solve the nonlinear complexities of the PDE.

Methodology

The research relies on the experimental system of the research, according to which several types of machine learning (neural networks, support vector machines, decision trees, convolutional neural networks, and residual networks) are applied to non-linear problems of the PDE. Assessment of the effectiveness of these models is done by comparison between the machine learning solutions and the conventional numerical methods using the synthetic and actual data. Accuracy, compute efficiency, and scalability are the most problematic performance measurements.

Findings

It is also discovered in the paper that deep learning architectures or models, and more specifically, convolutional neural networks (CNNs) and residual networks (ResNets), are orders of magnitude more effective as compared to their more traditional numerical counterparts in terms of accuracy and their own computational abilities. These models also contain high levels of scalability regulations through the employment of high-dimensional PDEs when contrasted with the classical methods. The results suggest that machine learning techniques, and specifically the deep-learning technique, possess colossal opportunities for the more effective resolution of the nonlinear PDEs.

Implications

The study shows that the science of computational science can be changed with the assistance of machine learning approaches by providing a more accurate and faster response to the large nonlinear Eulerian equations. Also, machine learning would be arguably much cheaper to solve specific problems in high-dimensional problems and therefore would be valuable in different applications, like in the field of financial engineering, climate tests, and fluid physics.