Efficient Neural Operator Decomposition via Tensor Product Quadrature

Recent studies have shown that the operator eigenvalue/singular value decomposition (EVD/SVD) can be efficiently achieved via neural network-based optimization; however, the function quadrature within the objective is computationally intensive, especially in high dimensions. Thus, mini-batch Monte Carlo sampling is used to obtain unbiased quadrature estimates, yet higher accuracy relies on larger batch size, imposing a computation-accuracy trade-off that restricts the efficiency of current methods in $N\ge 2$ dimensions. We propose a tensor product quadrature (TPQ) for operator decomposition, where each spectral function is parameterized by a variable-separable neural network, and the $N$-D quadrature is decomposed into high-accuracy and low-cost 1D quadratures followed by tensor product assembly, which largely bypasses the computation-accuracy trade-off. We further develop TPQ-compatible treatments for sampling, gradient mask, boundary constraints, and non-separable potentials using explicit decomposition or parametric pre-training. Numerical experiments on the Schr\"odinger equation (2D harmonic oscillator, 2D infinite well, 2D and 3D hydrogen atom), neural tangent kernel (NTK) decomposition, and up to 5D integral kernel operator SVD validate the TPQ method. Notably, the TPQ consistently improves the efficacy of neural operator EVD/SVD baselines, including spectral inference network (SpIN), neural eigenfunctions (NeuralEF), and nested low-rank approximation (NeuralSVD).