Value Problem

The "value problem," in the context of scientific computing, centers on efficiently and accurately solving differential equations, particularly boundary and initial value problems, which model numerous physical phenomena. Current research heavily utilizes neural networks, including physics-informed neural networks (PINNs), deep equilibrium models (DEQs), and graph neural networks (GNNs), often coupled with techniques like boundary integral equations or probabilistic numerics, to improve solution speed and accuracy, especially for complex or high-dimensional systems. These advancements offer significant potential for accelerating scientific simulations and engineering design across diverse fields, from fluid dynamics and materials science to autonomous driving safety analysis and healthcare applications.