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.

Stability analysis of hierarchical tensor methods for time-dependent PDEs

“…Such approximations allow us to significantly reduce the number of degrees of freedom in the representation of the solution tensor u(t), while maintaining accuracy. Low-rank tensor approximations of (2) can be constructed by using, e.g., rank-constrained temporal integration [34,28,13] on a (smooth) tensor manifold with constant rank [48]. Alternatively, one can utilize the fully discrete scheme (3) followed by a rank-reduction operation.…”

The numerical approximation of nonlinear functionals and functional differential equations

“…We also provided sufficient conditions for consistency, stability and convergence of functional approximation schemes to compute the solution of FDEs, thus extending the well-known Lax-Richtmyer theorem from PDEs to FDEs. As we suggested in [69], these results open the possibility to utilize techniques for highdimensional model representation such as deep neural networks [52,53,79] and numerical tensor methods [17,3,55,7,59,37] to represent nonlinear functionals and compute approximate solutions to functional differential equations. We conclude by emphasizing that the results we obtained in this paper can be extended to real-or complex-valued functionals in compact Banach spaces (see, e.g., [33,65]).…”

Spectral methods for nonlinear functionals and functional differential equations

“…The main results are Theorem 7.1 and Theorem 8.3, which are based on the Trotter-Kato approximation theorem for abstract evolution equations in Banach spaces. The results presented in this paper open the possibility to utilize techniques for high-dimensional function representation such as deep neural networks [73,74,104] and numerical tensor methods [5,10,[24][25][26]55,76,76,82] to approximate nonlinear functionals in terms of high-dimensional functions, and to compute approximate solutions of functional differential equations by solving high-dimensional PDEs.…”

“…The series (24) converges in the L 2 p ([0, 2π]) sense, and also pointwise since η is continuous [50]. A substitution of ( 24) into (20) yields…”