The field of scientific machine learning is rapidly advancing, with a particular focus on developing innovative methods for solving partial differential equations (PDEs) on complex domains. Recent research has made significant progress in overcoming the challenges associated with approximating functions and their derivatives on curved geometries, leveraging techniques such as deep neural networks on manifolds and neural operator learning. These advancements have far-reaching implications for various applications, including fluid flow, structural component deformation, and wave scattering problems. Notably, the development of novel neural network architectures, such as the Reduced Basis Neural Operator and the Neural Diffeomorphic-Neural Operator, has enabled efficient and accurate solutions to PDEs on complex geometries. Furthermore, the integration of physics-informed neural networks with other methods, such as the Petrov-Galerkin formulation and Fourier embeddings, has led to improved performance and robustness in solving PDEs. Some noteworthy papers in this regard include: Expressive Power of Deep Networks on Manifolds, which establishes a simultaneous approximation theory for deep neural networks on manifolds. ReBaNO: Reduced Basis Neural Operator, which proposes a novel data-lean operator learning algorithm for solving PDEs with multiple distinct inputs. FEDONet : Fourier-Embedded DeepONet, which introduces Fourier-embedded trunk networks within the DeepONet architecture for spectrally accurate operator learning.