Machine learning has become an increasingly important tool in the study of partial differential equations (PDEs). PDEs are fundamental in modeling a wide range of physical phenomena, from fluid dynamics and heat transfer to quantum mechanics and electromagnetics. However, solving PDEs can be a challenging task as they usually do not have a analytical solution. While a lot of the numerical methods have been developed to solve PDEs, they are often costly to run. This is where machine learning can help.

1.Machine learning provides a way to approximate the solution of a PDE without having to solve it explicitly. This is done by training a machine learning model on a set of data, typically generated by solving the PDE numerically or through experiment. The trained model can then be used to make predictions for new inputs, allowing us to directly estimate the solution of the PDE at locations where we don't have explicit solutions.

2. Machine learning techniques such as neural networks can be used to discover new PDEs or to learn the underlying physics of a system directly from data. This can be useful when the underlying PDE is unknown or when we have limited knowledge of the system. 

3. Machine learning can be used to assist classical numerical solver. Numerical solvers are generally sensitive to the discretization and too coarse discretization will lead to large error due to the under-resolved physics. Machine learning enables running numerical simulations on coarser discretization by learning to interpolate and correct the under-resolved physics for the numerical solver and thus accelerate the calculation process.