# Quantum-Inspired Tensor Neural Networks for Partial Differential Equations

arXiv: https://arxiv.org/abs/2208.02235 · HTML: https://mlsanity.com/paper/2208.02235/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 34 references that are in the map.
- First posted on arXiv 3 Aug 2022.

## Builds on (most cited first, 18 of 18)

- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Predicting Parameters in Deep Learning](https://mlsanity.com/paper/1306.0543/) (2013)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [Tree Tensor Networks for Generative Modeling](https://mlsanity.com/paper/1901.02217/) (2019)
- [Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations](https://mlsanity.com/paper/1804.07010/) (2018)
- [Learning Relevant Features of Data with Multi-scale Tensor Networks](https://mlsanity.com/paper/1801.00315/) (2017)
- [Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure](https://mlsanity.com/paper/1710.04833/) (2017)
- [From probabilistic graphical models to generalized tensor networks for supervised learning](https://mlsanity.com/paper/1806.05964/) (2018)
- [Generative Tensor Network Classification Model for Supervised Machine Learning](https://mlsanity.com/paper/1903.10742/) (2019)
- [Modeling Sequences with Quantum States: A Look Under the Hood](https://mlsanity.com/paper/1910.07425/) (2019)
- [Matrix Product State Based Quantum Classifier](https://mlsanity.com/paper/1905.01426/) (2019)
- [Don't overfit the history -- Recursive time series data augmentation](https://mlsanity.com/paper/2207.02891/) (2022)

## Built on by (most cited first, 9 of 9)

- [CompactifAI: Extreme Compression of Large Language Models using Quantum-Inspired Tensor Networks](https://mlsanity.com/paper/2401.14109/) (2024)
- [Quantum-Inspired Machine Learning: a Survey](https://mlsanity.com/paper/2308.11269/) (2023)
- [Quantum Large Language Models via Tensor Network Disentanglers](https://mlsanity.com/paper/2410.17397/) (2024)
- [Financial Risk Management on a Neutral Atom Quantum Processor](https://mlsanity.com/paper/2212.03223/) (2022)
- [Boosting Defect Detection in Manufacturing using Tensor Convolutional Neural Networks](https://mlsanity.com/paper/2401.01373/) (2023)
- [Quantum-Inspired Tensor Neural Networks for Option Pricing](https://mlsanity.com/paper/2212.14076/) (2022)
- [Application of Tensor Neural Networks to Pricing Bermudan Swaptions](https://mlsanity.com/paper/2304.09750/) (2023)
- [Efficient Finite Initialization with Partial Norms for Tensorized Neural Networks and Tensor Networks Algorithms](https://mlsanity.com/paper/2309.06577/) (2023)
- [Parameter-Minimal Neural DE Solvers via Horner Polynomials](https://mlsanity.com/paper/2602.14737/) (2026)

## Abstract (arXiv)

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.