The recent developments in the research area of partial differential equations (PDEs) and their applications have shown a significant shift towards leveraging machine learning techniques for more efficient and accurate solutions. The field is increasingly adopting hybrid models that integrate neural networks with traditional numerical methods, aiming to enhance computational efficiency and reduce the dependency on extensive labeled data. This trend is evident in the use of PDE-preserved coarse correction networks, autoregressive neural emulators, and differentiable transport equation models, all of which demonstrate advancements in handling complex spatiotemporal dynamics and high-dimensional problems. Notably, the integration of machine learning with physical models is not only improving prediction accuracy but also enabling the discovery of new physical relationships from data, a process often referred to as 'discovering physics'. Additionally, the field is witnessing a rise in the use of advanced computational tools like JAX for accelerated and differentiable simulations, which are crucial for large-scale neural network integration. The emphasis on interpretability, generalizability, and scalability in these models underscores a broader movement towards more robust and versatile solutions in PDE research. Furthermore, the application of these advancements spans across various domains, including fluid dynamics, population balance equations, and uncertainty quantification, highlighting the interdisciplinary impact of these innovations.

Noteworthy papers include 'P$^2$C$^2$Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics,' which introduces a novel approach to solving PDEs on coarse grids with high accuracy, and 'Modern, Efficient, and Differentiable Transport Equation Models using JAX: Applications to Population Balance Equations,' which showcases significant speed improvements in PBE simulations through the use of JAX.