The field of partial differential equations (PDEs) is witnessing a significant shift towards leveraging machine learning techniques to improve the accuracy and efficiency of solving these equations. Researchers are exploring innovative methods that combine traditional numerical schemes with deep learning approaches to tackle complex PDE problems. One notable direction is the development of stochastic solvers that incorporate neural networks to provide rapid and precise global solutions and gradient approximations. Another area of focus is the use of deep neural networks to compute solutions of evolutionary PDEs in high dimensions, taking advantage of the expression power of DNNs and the simplicity of tangent bundle approximation. Furthermore, there is a growing interest in analyzing the theoretical foundations of these methods, including error analysis and generalization bounds, to ensure their reliability and efficiency. Noteworthy papers in this area include:

• WoSNN, which proposes a novel stochastic solver that integrates machine learning techniques with the Walk-on-Spheres method, achieving accurate field estimations and reducing errors by around 75%.
• The Deep Tangent Bundle method, which develops a numerical framework for computing solutions of evolutionary PDEs in high dimensions using deep neural networks.
• Error analysis for learning the time-stepping operator of evolutionary PDEs, which provides a rigorous theoretical framework for analyzing the approximation of these operators using feedforward neural networks.