My research now focuses on two aspects, forward problems and inverse problems. The current idea is to use machine learning methods to solve these problems from different perspectives.

Forward problems:

1. Parareal-in-time algorithm (parareal algorithm)
   Here we consider to solve the PDEs with time parallel algorithms, the most famous one would be the parareal algorithm. The basic idea is to construct a coarse solver and a fine solver and then we will firstly make a prediction using the coarse solver and the make a correction using the fine solver. I now focus on developing coarse solvers based on different methods such as deep learning methods, POD, and so on.

2. Physics-informed neural networks (PINNs)
   PINNs is the most advanced one to solve PDEs based on the deep learning methods. The basic idea is to replace the original underlying solution with a neural network and optimize a physics-informed loss to approximate the solution. I now focus on develop adaptive sampling strategies to discrete the loss functional to accelerate the convergence.

3. Deep operator learning (DeepONet)
   The deep operator learning aims to construct a neural network based operator to map functions in the input space to the functions in the output space. The current models include DeepOnet, FNO, GNO etc. I now focus on decreasing the computational cost for generating the training dataset based on a fine solver.

Inverse problems:

1. Bayesian inference
   The Bayesian inference is going to find the posterior distribution using the observation data as the likelihood function by defining a proper prior distribution at first. The current idea is to replace the original expensive forward model with cheap surrogates and then run sampling methods to do inference. I am currently working on improving the accuracy of the surrogates and establish rigorous error bounds.

2. Sci-ML methods for inverse problems
   Recently, many scientific machine learning methods have been proposed to model the inverse problems, including regularization theory. I now want to develop direct sampling methods to explore the posterior distribution, including VAE-based models and operator-learning based models.

For future research plans, I am going to learn more theory about the probability.