Neural Network Approximation to Piecewise Constant Functions and its Applications to Hyperbolic Conservation Laws
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In this talk, we will present our recent result on approximation properties of neural networks (NNs) to piecewise constant functions with unknown interface, in terms of NN architecture and error bound. This result shows that NNs are superior to piecewise polynomials on fixed meshes when approximating discontinuous functions with unknown interface. We will then describe a space-time least-squares neural network (LSNN) method for solving nonlinear scalar hyperbolic conservation laws, that employs a physics-preserved discrete divergence operator. The method is free of any penalization such as the entropy, total variation, and/or artificial viscosity, etc., and it sharply capture shock without oscillation, overshooting, or smearing. If time permits, we will describe our newly developed training algorithm.
