Dynamically orthogonal tensor methods for high-dimensional nonlinear PDEs

Abstract

We develop new dynamically orthogonal tensor methods to approximate multivariate functions and the solution of high-dimensional time-dependent nonlinear partial differential equations (PDEs). The key idea relies on a hierarchical decomposition of the approximation space obtained by splitting the independent variables of the problem into disjoint subsets. This process, which can be conveniently be visualized in terms of binary trees, yields series expansions analogous to the classical Tensor-Train and Hierarchical Tucker tensor formats. By enforcing dynamic orthogonality conditions at each level of binary tree, we obtain coupled evolution equations for the modes spanning each subspace within the hierarchical decomposition. This allows us to effectively compute the solution to high-dimensional time-dependent nonlinear PDEs on tensor manifolds of constant rank, with no need for rank reduction methods. We also propose new algorithms for dynamic addition and removal of modes within each subspace. Numerical examples are presented and discussed for high-dimensional hyperbolic and parabolic PDEs in bounded domains.

39 pages, 19 figures

References in corpus (6)

• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
• Adaptive Leja sparse grid constructions for stochastic collocation and high-dimensional approximation
• The Numerical Approximation of Nonlinear Functionals and Functional Differential Equations
• Optimal randomized multilevel algorithms for infinite-dimensional integration on function spaces with ANOVA-type decomposition
• Parallel numerical tensor methods for high-dimensional PDEs

Cited by in corpus (8)

• A Low Rank Tensor Representation of Linear Transport and Nonlinear Vlasov Solutions and Their Associated Flow Maps
• Tensor methods for the Boltzmann-BGK equation
• Functional Tucker approximation using Chebyshev interpolation
• Dynamic tensor approximation of high-dimensional nonlinear PDEs
• Stability analysis of hierarchical tensor methods for time-dependent PDEs
• Tensor rank reduction via coordinate flows
• Fast global spectral methods for three-dimensional partial differential equations
• Approximation in the extended functional tensor train format