The recent advancements in the field of neural networks and their applications to differential equations and optimization problems have shown significant progress. Researchers are focusing on developing more efficient and scalable algorithms for computing higher-order derivatives, which is crucial for physics-informed neural networks and other applications. Novel frameworks like Forward-Backward Stochastic Jump Neural Networks are being introduced to solve complex equations more efficiently by reducing the total number of parameters. Compositional learning algorithms for constrained dynamical systems, such as Neural Port-Hamiltonian Differential Algebraic Equations, are addressing the challenges posed by algebraic constraints, leading to improved prediction accuracy and constraint satisfaction. In the realm of optimization, the scalability of neural network surrogates is being enhanced through innovative formulations and GPU acceleration, enabling the handling of larger models within acceptable time frames. Additionally, semi-implicit neural ODEs are being developed to tackle stiff problems more effectively, offering enhanced stability and computational efficiency. Real-time simulation of complex biological systems is also seeing advancements with the integration of graph neural networks and physical constraints, enabling high-speed predictions with remarkable generalization. Lastly, new frameworks for neural PDE surrogates are being proposed that predict temporal derivatives instead of states, offering greater flexibility and accuracy. These developments collectively indicate a shift towards more efficient, interpretable, and scalable solutions in the application of neural networks to complex mathematical and physical problems.