Synthetic dimensions (SDs) opened the door for exploring previously inaccessible phenomena in high-dimensional space. However, construction of synthetic lattices with desired coupling properties is a challenging and unintuitive task. Here, we use deep learning artificial neural networks (ANNs) to construct lattices in real space with a predesigned spectrum of mode eigenvalues, and thus to validly design the dynamics in synthetic mode dimensions. By employing judiciously chosen perturbations (wiggling of waveguides at desired frequencies), we show resonant mode coupling and tailored dynamics in SDs. Two distinct examples are illustrated: one features uniform synthetic mode coupling, and the other showcases the edge defects that allow for tailored light transport and confinement. Furthermore, we demonstrate morphing of light into a topologically protected edge mode with modified Su–Schrieffer–Heeger photonic lattices. Such an ANN-assisted construction of SDs may advance toward “utopian networks,” opening new avenues for fundamental research beyond geometric limitations as well as for applications in mode lasing, optical switching, and communication technologies.

Optical bound states in the continuum (BICs) have recently stimulated a research boom, accompanied by demonstrations of abundant exotic phenomena and applications. With ultrahigh quality (Q) factors, optical BICs have powerful abilities to trap light in optical structures from the continuum of propagation waves in free space. Besides the high Q factors enabled by the confined properties, many hidden topological characteristics were discovered in optical BICs. Especially in periodic structures with well-defined wave vectors, optical BICs were discovered to carry topological charges in momentum space, underlying many unique physical properties. Both high Q factors and topological vortex configurations in momentum space enabled by BICs bring new degrees of freedom to modulate light. BICs have enabled many novel discoveries in light–matter interactions and spin–orbit interactions of light, and BIC applications in lasing and sensing have also been well explored with many advantages. In this paper, we review recent developments of optical BICs in periodic structures, including the physical mechanisms of BICs, explored effects enabled by BICs, and applications of BICs. In the outlook part, we provide a perspective on future developments for BICs.