Quantum-Inspired Tensor Neural Networks for Partial Differential Equations
This addresses computational bottlenecks for researchers and practitioners in fields like finance and engineering who solve PDEs, though it is incremental as it builds on existing tensor network ideas.
The paper tackled the training time and memory constraints of deep learning methods for solving high-dimensional partial differential equations (PDEs) by implementing Tensor Neural Networks (TNN), a quantum-inspired architecture, and demonstrated that TNN achieves the same accuracy as classical Dense Neural Networks (DNN) with significant parameter savings and faster training.
Partial Differential Equations (PDEs) are used to model a variety of dynamical systems in science and engineering. Recent advances in deep learning have enabled us to solve them in a higher dimension by addressing the curse of dimensionality in new ways. However, deep learning methods are constrained by training time and memory. To tackle these shortcomings, we implement Tensor Neural Networks (TNN), a quantum-inspired neural network architecture that leverages Tensor Network ideas to improve upon deep learning approaches. We demonstrate that TNN provide significant parameter savings while attaining the same accuracy as compared to the classical Dense Neural Network (DNN). In addition, we also show how TNN can be trained faster than DNN for the same accuracy. We benchmark TNN by applying them to solve parabolic PDEs, specifically the Black-Scholes-Barenblatt equation, widely used in financial pricing theory, empirically showing the advantages of TNN over DNN. Further examples, such as the Hamilton-Jacobi-Bellman equation, are also discussed.