The field of neural operators for partial differential equations (PDEs) is rapidly advancing, with a focus on developing more accurate and efficient models for complex problems. Recent research has highlighted the limitations of current neural operator architectures in capturing transport dynamics governed by discontinuities, prompting the development of new methods that can better preserve physical features such as shock waves.

One of the key directions in this field is the development of novel neural operator architectures that can effectively handle high-frequency components and non-linear dynamics. This includes the use of Proper Orthogonal Decomposition (POD) and Fourier spectral methods to construct more efficient and accurate integral kernels.

Another area of focus is the development of models that can discover evolution operators for multi-physics-agnostic prediction, allowing for more efficient and accurate prediction of dynamic systems governed by unknown temporal PDEs. This has led to the development of new frameworks that decouple dynamics estimation from state prediction, enabling more effective use of data and improved generalization performance.

Notable papers in this area include:

• PODNO, which introduces a new neural operator architecture that leverages POD to improve accuracy and efficiency for high-frequency problems.
• DISCO, which proposes a framework for discovering evolution operators from short trajectories and achieves state-of-the-art performance on diverse physics datasets.
• Temporal Neural Operator, which develops an efficient neural operator for spatio-temporal operator learning and demonstrates long-range temporal extrapolation capabilities and robustness to error accumulation.
• FourierSpecNet, which integrates the Fourier spectral method with deep learning to approximate the collision operator in Fourier space efficiently and achieves competitive accuracy while reducing computational cost.