Abstract: Analytically solving complex or large-scale differential equations is often difficult or even impossible, making numerical integration methods indispensable. However, as all numerical ...
A $1 million prize awaits anyone who can show where the math of fluid flow breaks down. With specially trained AI systems, ...
Accurately modeling steady-state two-phase flow is critical for the design and operation of systems in the oil and gas industry; however, traditional models often struggle to adapt to specific field ...
Tessellations aren’t just eye-catching patterns—they can be used to crack complex mathematical problems. By repeatedly ...
Leevan Ling and Manfred R Trummer. Adaptive multiquadric collocation for boundary layer problems. Journal of Computational and Applied Mathematics, 188(2):265–282, 2006. Richard Baltensperger and ...
Abstract: Physics-Informed Neural Networks (PINNs) have recently emerged as a powerful method for solving differential equations by leveraging machine learning techniques. However, while neural ...
The parabolic equation (PE) serves as a fundamental methodology for modeling underwater acoustic propagation. The computational efficiency of this approach derives from the far-field approximation of ...
To derive Newton's method, it is convenient to start with a Taylor series expansion of the residual function: $$ R(u_i) = R(u_{i-1}) + \frac{\partial \textbf{R ...
A modular framework implementing Physics-Informed Neural Networks (PINNs) with Gradient Normalization for solving differential equations including Navier-Stokes equations for crystal growth modeling.
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