Theory and computation of Wasserstein geometric flows with application to time-dependent Schrodinger equation
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Wu, Hao
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Abstract
We focus on the systematic study of a novel computational framework to the Wasserstein geometric flows, which describe time evolution of probability density functions on the infinite-dimensional Wasserstein manifold. In particular, Wasserstein gradient flows (WGFs) and Wasserstein Hamiltonian flows (WHFs), are the main examples used throughout this research—they have many applications in real-world physics systems and more recently in deep learning problems such as generative models.
The main feature of our computational framework is to use deep neural networks, to parameterize the push-forward maps such that they can push a simple reference density to the ones solving the WGFs or WHFs. This approach essentially reduces these flows defined on infinite-dimensional Wasserstein manifold to finite-dimensional dynamical systems of the parameters. These new dynamical systems are parameterizations of the WGFs and WHFs, which we call PWGFs and PWHFs for short. By leveraging a relaxed pullback Wasserstein metric on the parameter space, we can develop effective numerical methods to approximate the solutions of these flows.
For WGFs, we show that our proposed PWGF scheme can be applied to WGF with general energy functional. Moreover, our scheme does not require any spatial discretization and thus is scalable to cases where the space dimensions of the problems are high. Our approach only requires solving standard least
squares problems in each time step, hence is training free. With these features, PWGF demonstrates promising computational efficiency and accuracy on a variety of WGF examples, as shown in our numerical experiments.
For WHFs, we adopt the similar idea but apply it to the more challenging Hamiltonian systems on Wasserstein manifolds. To preserve the Hamiltonian, we employ a symplectic numerical scheme to solve the PWHF, where a fixed-point iteration scheme is used to solve the implicit update equation of the model parameter. Similar to PWGF, PWHF is training free and thus avoids the issues of nonconvex optimization algorithms. We also present the connection between the Lagrangian and Eulerian perspectives of the original flows using PWHF. Approximation error analysis
and a number of numerical examples are also provided using PWHF. Furthermore, we also consider the Schr\"odinger equations (SEs), and show how to use PWHF to solve them.
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Date
2023-07-26
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Dissertation