Numerical computation of Partial Differential Equations (PDE) is by no doubt very important in science and technology. Unfortunately, computations with classical numerical methods with finite element/volume/difference methods are computationally very heavy and often done on big clusters. Important applications are simulations of fluids or structural systems. The classical methods also suffer under the curse of dimensionality, meaning in practice that PDEs with more than 4 space variables are usually impossible to solve numerically, even on clusters, due to the computational complexity.
Deep learning has revolutionized computer vision. When it comes to numerical PDE the revolution is not yet here. Still, the last 2 years the research in this field has increased from almost nothing to quite a number of interesting papers. In the preprint
(Summary by Two Minute Papers: https://youtu.be/iOWamCtnwTc)
the authors use a hybrid finite difference / deep learning approach to generate smoke in computer graphics by solving the Incompressible Euler Equation from fluid dynamics. In the preprint
the authors use a connection between PDE and Backward Stochastic Differential Equations (BSDE). The PDE problem can be formulated as a BSDE problem and the BSDE can be solved approximately with Deep Neural Networks. A number of PDEs with 100 space variables (!) were solved in the paper.
At Syntronic we have modified and evaluated the latter method on problems in 4 and 6 dimension. By solving the Hamilton-Jacobi-Bellman PDE we can generate a good approximation of the optimal feedback control of a non-linear stochastic control problem. As applications, we consider the feedback control of single and double inverted pendulums, in real time (!). We also have some preliminary results on nonlinear feedback control of a vehicle, modelled with the bicycle model. The major part of the work is from the master thesis project of Kristoffer Andersson, Chalmers. The report will be available soon under the title "Approximate stochastic control based on deep learning and backward stochastic differential equations".
Parts of the talks will be somewhat technical and for the Meetup to be of interesting to you, it is good if you know why it is important to solve PDEs and have some mathematical maturity.
Schedule for the evening:
18:00 – 18:30: Food and mingle
18:30 – 19:00: A literature overview of deep learning for solving PDE
19:00 – 19:10: Short break and time for questions
19:10 – 19:40: Approximate stochastic control using deep learning and stochastic control theory
19:40 – 20:30: Mingle and discussion
Speaker of both talks is Adam Andersson, PhD, team leader at Syntronic.