Mon, Mar 30

Algebra Seminar

Dmytro Matvieievskyi, UMass Amherst

TBA

4:00 PM, 250 Mathematics Building

TBA

Fri, Apr 10

Applied Mathematics Seminar

Yulong Lu (U Minnesota)

Title: TBD

4:00 PM, Room: TBD

Fri, Apr 10

Topology and Geometry Seminar

Roberta Shapiro (University of Michigan)

TBA

4:00 PM, 122 Mathematics Building

TBA

Wed, Apr 15

Analysis Seminar

Rizwanur Khan, University of Texas at Dallas

TBA

4:00 PM, 250 Mathematics building

TBA

Fri, Apr 17

Applied Mathematics Seminar

Lili Ju (U of South Carolina)

Transferable Neural Networks for Partial Differential Equations

4:00 PM, MATH250

Transfer learning for partial differential equations (PDEs) aims to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information about the target PDEs such as its formulation and/or data of its solution for pre-training. In this work, we propose to design transferable neural feature spaces for the shallow neural networks from purely function approximation perspectives without using PDE information. The construction of the feature space involves the re-parameterization of the hidden neurons and uses auxiliary functions to tune the resulting feature space. Theoretical analysis shows the high quality of the produced feature space, i.e., uniformly distributed neurons. We use the proposed feature space as the predetermined feature space of a random feature model and use existing least squares solvers to obtain the weights of the output layer. Extensive numerical experiments verify the outstanding performance of our method, including significantly improved transferability, e.g., using the same feature space for various PDEs with different domains and boundary conditions, and the superior accuracy, e.g., several orders of magnitude smaller mean squared error than the state-of-the-art methods. Finally, we discuss ongoing and future research topics along this direction.