Machine learning neural networks method solving partial differential equations

In recent years, rapid growth in theory and applications of machine learning have occurred. This growth has been driven by advances in deep learning, which has achieved tremendous success such as large-scale image classification. At the same time, exciting new work is exploring connections between classical fields of mathematics, such as partial differential equations (PDEs) and machine learning. Optimal control of ordinary differential equations has been connected recently with the stability of ResNet neural networks. One benefit of deep learning is that once trained they are computationally fast to apply. Hence, this offers new ways to tackle computationally challenging problems, like the numerical solution of high-dimensional and non-linear PDEs.

In the spring semester, we will start with a 4 weeks session to learn the basics of neural networks. Solving ordinary differential equations system, i.e. Lorenz System, will be considered as a model problem to learn how to build up feed forward networks with weight matrices and biases, chain rule of backward propagation and applying gradient descent method to search the minimum solution. Then we will explore and read recently published research articles, especially those related to physics-based models with data-driven components constrained to conservation principles originating from the physical laws. Toward the second half of the semester, we will apply neural networks and explore ideas to solve linear and nonlinear partial differential equations, i.e. Burgers equation, which serves as the model problem for applications in fluid dynamics of compressible flow. Algorithm and method implementations with MatLab (or Python with TensorFlow) are expected. Extension or extra effort into the summer may lead to one paper publication in an undergraduate level journal.

For more information contact Changxin Qu (cxqiu@iastate.edu)

People:

  • Changxin Qiu (Postdoc)

  • Jue Yan (Faculty)

  • Jacob Beattie (undergrad)

  • Anders Lie (undergrad)

  • Zhehao Xing (undergrad)

  • Zizheng Yang (undergrad)

Pre-requisites:

  • Experience with programming (e.g. Matlab, Python) is desirable, but not necessary.

  • Experience with Elementary Differential Equations(Math 266/267) is desirable, but not necessary.