Functional tensor train neural network for solving high-dimensional PDEs

TL;DR

The paper introduces a functional tensor train neural network (FTTNN) for solving high-dimensional PDEs on irregular domains, using a functional tensor train (FTT) representation with TT-core functions parameterized by neural networks. It defines the functional tensor train rank and establishes an approximation property showing that, for any $\epsilon>0$, a suitable FTT-rank and neural-network cores can approximate the target solution in $L^2(\Omega)$ to within $\epsilon$. Training uses a physics-informed loss that leverages TT structure to reduce high-dimensional integrals to one-dimensional Gauss quadratures, avoiding sampling errors. Extensive numerical experiments across Poisson, Helmholtz, and Schrödinger problems demonstrate that FTTNN often outperforms discrete TT methods and PINN/DRM baselines, especially on irregular domains and higher dimensions, indicating strong potential for scalable, accurate high-dimensional PDE solvers.

Abstract

Discrete tensor train decomposition is widely employed to mitigate the curse of dimensionality in solving high-dimensional PDEs through traditional methods. However, the direct application of the tensor train method typically requires uniform grids of regular domains, which limits its application on non-uniform grids or irregular domains. To address the limitation, we develop a functional tensor train neural network (FTTNN) for solving high-dimensional PDEs, which can represent PDE solutions on non-uniform grids or irregular domains. An essential ingredient of our approach is to represent the PDE solutions by the functional tensor train format whose TT-core functions are approximated by neural networks. To give the functional tensor train representation, we propose and study functional tensor train rank and employ it into a physics-informed loss function for training. Because of tensor train representation, the resulting high-dimensional integral in the loss function can be computed via one-dimensional integrals by Gauss quadrature rules. Numerical examples including high-dimensional PDEs on regular or irregular domains are presented to demonstrate that the performance of the proposed FTTNN is better than that of Physics Informed Neural Networks (PINN).

Figures (9)

• Figure 1: Solutions of three-dimensional Poisson equations on L shape domain at $x_3=0.5$, test problem 2.
• Figure 2: Relative error vs epoch ($d=3, r=2$), test problem 3:
• Figure 3: Solutions of three-dimensional Poisson equations on regular domains at $x_3=0.1$, test problem 3.
• Figure 4: Relative error vs epoch ($d=5, r=3$), test problem \ref{['h1']}
• Figure 5: Solutions of five-dimensional Helmoltz equations at $x_3=x_4=x_5=0.1$, test problem \ref{['h1']}.

Theorems & Definitions (8)

• Theorem 2.1: Discrete tensor train decomposition
• Definition 2.2: Tensor function
• Definition 2.3: Sampled tensor set
• Definition 2.4: Functional tensor train rank
• Proposition 2.5
• Theorem 2.6: Functional tensor train decomposition
• Theorem 3.1: Approximation error
• Proof 1