The field of solving partial differential equations (PDEs) using neural methods is rapidly advancing, with a focus on improving the accuracy, stability, and efficiency of these models. Recent developments have highlighted the importance of incorporating local spatial features and leveraging techniques such as denoising and diffusion models to improve the performance of neural PDE solvers. Another area of focus is on developing more efficient and stable methods for solving PDEs, including the use of continuous data assimilation and generative latent neural models. Overall, these advances are paving the way for more accurate and reliable simulations of complex systems.

Noteworthy papers include:

• Enhancing Fourier Neural Operators with Local Spatial Features, which introduces a hybrid architecture that combines the strengths of Fourier neural operators and convolutional neural networks to improve the representational capacity of neural PDE solvers.
• Thermalizer: Stable autoregressive neural emulation of spatiotemporal chaos, which proposes a novel method for stabilizing autoregressive emulator rollouts using diffusion models, allowing for more accurate and long-term predictions of complex systems.
• Generative Latent Neural PDE Solver using Flow Matching, which presents a latent diffusion model for PDE simulation that reduces computational costs and improves accuracy and stability.