The field of partial differential equations (PDEs) is witnessing significant advancements in solver discovery and operator learning. Researchers are exploring the integration of classical numerical methods with modern deep learning components to develop flexible and efficient solvers. The use of neural operators, dynamic mode decomposition, and neural Hamiltonian operators is becoming increasingly popular for solving time-dependent nonlinear PDEs. These approaches enable the automated discovery of optimal solvers, balance performance criteria such as accuracy and speed, and provide a principled foundation for solver selection. Furthermore, the development of continuous-time operator networks is allowing for the accurate learning of solution operators from sparse and irregular data, making it possible to query predictions on unseen meshes and time steps without retraining or interpolation. Noteworthy papers include:

• A paper presenting a general and scalable framework for the automated discovery of optimal meta-solvers, which consistently outperform conventional iterative methods.
• A study introducing a neural operator based on dynamic mode decomposition, which achieves high reconstruction accuracy and reduces computational costs compared to traditional numerical methods.
• A paper introducing a Neural Hamiltonian Operator for solving stochastic control problems in high dimensions, which situates the deep FBSDE method within the rigorous language of statistical inference.
• A work presenting NCDE-DeepONet, a continuous-time operator network that enables input-resolution-independent representation and output-resolution-independent prediction, achieving almost instant solution prediction on various transient problems.