The field of scientific computing is witnessing significant advancements in the development of physics-informed neural networks and inverse problem solving. Recent research has focused on improving the accuracy and efficiency of these models, particularly in capturing complex physical phenomena and solving high-dimensional problems. Notably, the incorporation of symmetry and equivariance in neural network architectures has led to improved performance in modeling multistability and bifurcation phenomena. Additionally, the development of novel optimization techniques and adaptive sampling methods has enhanced the ability to solve inverse problems with increased accuracy and reduced computational cost.

Some noteworthy papers in this area include the introduction of Equivariant U-Shaped Neural Operators for the Cahn-Hilliard Phase-Field Model, which achieves accurate predictions across space and time by encoding symmetry and scale hierarchy. The Feynman-Kac-Flow approach has also been proposed for inference steering of conditional flow matching, enabling the generation of samples that meet precise requirements. Furthermore, the development of Neuro-Spectral Architectures for causal physics-informed networks has addressed issues of spectral bias and causality in standard PINNs, leading to improved performance in solving partial differential equations.