Solving parametric PDE problems with artificial neural networks
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The curse of dimensionality is commonly encountered in numerical partial differential equations (PDE), especially when uncertainties have to be modeled into the equations as random coefficients. However, very often the variability of physical quantities derived from PDE can be captured by a few features on the space of random coefficients. Based on such observation, we propose using neural-network, a technique gaining prominence in machine learning tasks, to parameterize the physical quantity of interest as a function of random input coefficients. The simplicity and accuracy of the approach are demonstrated through notable examples of PDEs in engineering and physics.
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Professor of Mathematics
Jianfeng Lu is an applied mathematician interested in mathematical analysis and algorithm development for problems from computational physics, theoretical chemistry, materials science and other related fields.More specifically, his current research focuses include:Electronic structure and many body problems; quantum molecular dynamics; multiscale modeling and analysis; rare events and sampling techniques.