Thesis ; Online: Applications of Deep Learning to Scientific Computing
2023
Abstract: Physics-informed neural networks (PINNs) have been widely used for the robust and accurate approximation of partial differential equations. In the present thesis, we provide upper bounds on the generalization error of PINNs approximating solutions to the ...
| Abstract | Physics-informed neural networks (PINNs) have been widely used for the robust and accurate approximation of partial differential equations. In the present thesis, we provide upper bounds on the generalization error of PINNs approximating solutions to the forward and inverse problems for PDEs. Specifically, we focus on a particular class of inverse problems, the so-called data assimilation or unique continuation problems. An abstract formalism is introduced, and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and the number of training samples. This abstract framework is illustrated with several examples of PDEs, and numerical examples validating the proposed theory are also presented. The derived estimates show two relevant facts: (1) PINNs require regularity of solutions to the underlying PDE to guarantee accurate approximation. Consequently, they may fail to approximate discontinuous solutions of PDEs, such as nonlinear hyperbolic equations. We then propose a novel variant of PINNs, termed weak PINNs (wPINNs), for accurate approximation of entropy solutions of scalar conservation laws. wPINNs are based on approximating the solution of a min-max optimization problem for a residual, defined in terms of Kruzhkov entropies, to determine parameters for the neural networks approximating the entropy solution as well as test functions. Moreover, (2) with a suitable quadrature rule, i.e., Monte Carlo quadrature, PINNs may potentially overcome the curse of dimensionality. Hence, we embrace physics-informed neural networks (PINNs) to solve the forward and inverse problems for a broad range of high-dimensional PDEs, including the radiative transfer equation and financial equations. We present a suite of numerical experiments demonstrating that PINNs provide very accurate solutions for both the forward and inverse problems at low computational cost without incurring the curse of dimensionality. In the final part of the thesis, we ... |
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| Keywords | Machine learning ; Computational science ; Differential equations ; info:eu-repo/classification/ddc/510 ; Mathematics |
| Subject code | 518 |
| Language | English |
| Publisher | ETH Zurich |
| Publishing country | ch |
| Document type | Thesis ; Online |
| Database | BASE - Bielefeld Academic Search Engine (life sciences selection) |
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