Functional tensor train neural network for solving high-dimensional PDEs

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).