# Solving high-dimensional parabolic PDEs using the tensor train format

arXiv: https://arxiv.org/abs/2102.11830 · HTML: https://mlsanity.com/paper/2102.11830/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 70 references that are in the map.
- First posted on arXiv 23 Feb 2021.

## Builds on (most cited first, 14 of 14)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space](https://mlsanity.com/paper/2005.05409/) (2020)
- [Deep splitting method for parabolic PDEs](https://mlsanity.com/paper/1907.03452/) (2019)
- [A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients](https://mlsanity.com/paper/1809.07321/) (2018)
- [High-Dimensional Stochastic Optimal Control using Continuous Tensor Decompositions](https://mlsanity.com/paper/1611.04706/) (2016)
- [Variational approach to rare event simulation using least-squares regression](https://mlsanity.com/paper/1901.09195/) (2019)
- [Solving non-linear Kolmogorov equations in large dimensions by using deep learning: a numerical comparison of discretization schemes](https://mlsanity.com/paper/2012.07747/) (2020)
- [Linear Hamilton Jacobi Bellman Equations in High Dimensions](https://mlsanity.com/paper/1404.1089/) (2014)
- [Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficients](https://mlsanity.com/paper/2101.01869/) (2021)
- [Approximative Policy Iteration for Exit Time Feedback Control Problems driven by Stochastic Differential Equations using Tensor Train format](https://mlsanity.com/paper/2010.04465/) (2020)

## Built on by (most cited first, 17 of 17)

- [Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inference](https://mlsanity.com/paper/2508.12511/) (2025)
- [Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems](https://mlsanity.com/paper/2112.03749/) (2021)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [Permutation Search of Tensor Network Structures via Local Sampling](https://mlsanity.com/paper/2206.06597/) (2022)
- [Alternating Local Enumeration (TnALE): Solving Tensor Network Structure Search with Fewer Evaluations](https://mlsanity.com/paper/2304.12875/) (2023)
- [Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation](https://mlsanity.com/paper/2302.05104/) (2023)
- [Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEs](https://mlsanity.com/paper/2505.03595/) (2025)
- [A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problems](https://mlsanity.com/paper/2412.09009/) (2024)
- [Cost-efficient Gaussian Tensor Network Embeddings for Tensor-structured Inputs](https://mlsanity.com/paper/2205.13163/) (2022)
- [From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEs](https://mlsanity.com/paper/2307.15496/) (2023)
- [tnGPS: Discovering Unknown Tensor Network Structure Search Algorithms via Large Language Models (LLMs)](https://mlsanity.com/paper/2402.02456/) (2024)
- [A block-sparse Tensor Train Format for sample-efficient high-dimensional Polynomial Regression](https://mlsanity.com/paper/2104.14255/) (2021)
- [Position: Tensor Networks are a Valuable Asset for Green AI](https://mlsanity.com/paper/2205.12961/) (2022)
- [Tensor Gaussian Processes: Efficient Solvers for Nonlinear PDEs](https://mlsanity.com/paper/2510.13772/) (2025)
- [Sampling from Boltzmann densities with physics informed low-rank formats](https://mlsanity.com/paper/2412.07637/) (2024)
- [Tensor Train Diffusion: Leveraging Low-Rank Structures for High-Dimensional Score-Based Sampling](https://mlsanity.com/paper/2607.06841/) (2026)
- [Accelerated Learning of High Dimensional Functions with a Tensor-Featured Training Network](https://mlsanity.com/paper/2608.10351/) (2026)

## Abstract (arXiv)

High-dimensional partial differential equations (PDEs) are ubiquitous in economics, science and engineering. However, their numerical treatment poses formidable challenges since traditional grid-based methods tend to be frustrated by the curse of dimensionality. In this paper, we argue that tensor trains provide an appealing approximation framework for parabolic PDEs: the combination of reformulations in terms of backward stochastic differential equations and regression-type methods in the tensor format holds the promise of leveraging latent low-rank structures enabling both compression and efficient computation. Following this paradigm, we develop novel iterative schemes, involving either explicit and fast or implicit and accurate updates. We demonstrate in a number of examples that our methods achieve a favorable trade-off between accuracy and computational efficiency in comparison with state-of-the-art neural network based approaches.