The recent advancements in the field of physics-informed neural networks (PINNs) and related methodologies have significantly enhanced the ability to model and solve complex partial differential equations (PDEs). A notable trend is the integration of adaptive and dynamic representations, such as conditional neural fields and Physics-Informed Gaussians, which allow for more flexible and accurate approximations of PDE solutions. These approaches address the limitations of traditional neural networks, such as spectral bias and fixed parameter positions, by incorporating trainable parameters that adjust during training. Additionally, unsupervised learning frameworks, like the Partition of Unity Physics-Informed Neural Networks, are being developed to identify and optimize spatial subdomains with specific governing physics, offering a novel way to tackle ill-posed inverse problems. Another innovative development is the use of transformer neural operators, which generalize well to unseen initial and boundary conditions without the need for extensive simulation data. This approach, exemplified by the physics-informed transformer neural operator (PINTO), demonstrates superior performance in handling various engineering applications. Furthermore, the application of PINNs to three-dimensional contact problems, with advancements in handling inequality constraints and boundary conditions, showcases the versatility and robustness of these methods in real-world scenarios.

Noteworthy papers include the introduction of conditional neural fields for reduced-order modeling, which combines parametric neural ODEs with a physics-informed learning objective, and the development of Physics-Informed Gaussians, which offer adaptive parametric mesh representations. The Partition of Unity Physics-Informed Neural Networks present an unsupervised framework for domain decomposition, while the physics-informed transformer neural operator (PINTO) efficiently generalizes to new initial and boundary conditions.