The current developments in the research area of neural networks and function approximation are notably advancing the field through innovative approaches and theoretical insights. A significant trend is the exploration of minimal-width neural networks, which aim to achieve universal approximation with fewer parameters, thereby enhancing computational efficiency and theoretical understanding. This is exemplified by the use of leaky ReLU activations and coding schemes that leverage standard $L^p$ results, leading to networks with minimal interior dimensions. Additionally, there is a growing focus on the robustness and interpretability of neural network architectures, particularly in safety-critical applications such as system theory and autonomous vehicles. This is highlighted by the development of quadratic convolutional neural networks trained via least squares, which offer analytic solutions and reduced training times. Furthermore, the integration of control theory with neural networks is yielding novel solutions for time-variant problems under uncertainty, enhancing both convergence speed and computing accuracy. These advancements not only deepen the theoretical foundations of neural networks but also pave the way for practical applications in complex, real-world scenarios.

Noteworthy papers include one that demonstrates the minimal width required for universal approximation using leaky ReLU activations, and another that introduces a continuous domain for function spaces, enabling computations in non-core-compact topological spaces.