mlsanityData to 7 Oct 2026

Solving high-dimensional parabolic PDEs using the tensor train format

arXiv 2102.11830 ↗First posted 23 Feb 2021stat.ML cs.LG cs.NAcited by 17

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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.

Key facts. Cited by 17 papers in the map (data through 7 Oct 2026). Builds on 14 of its 70 references that are in the map. First posted on arXiv 23 Feb 2021.

Builds on

14 of 70 references.

References · 14

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2016Densely Connected Convolutional Networkscited by 6,691
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2 more
  1. 2021Convergence of the Deep BSDE method for FBSDEs with non-Lipschitz coefficientscited by 7
  2. 2020Approximative Policy Iteration for Exit Time Feedback Control Problems driven by Stochastic Differential Equations using Tensor Train formatcited by 4

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Citing papers per year: 2021: 2; 2022: 3; 2023: 4; 2024: 3; 2025: 3; 2026: 2.

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  1. 2025Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inferencecited by 22
  2. 2021Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problemscited by 21
  3. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  4. 2022Permutation Search of Tensor Network Structures via Local Samplingcited by 13
  5. 2023Alternating Local Enumeration (TnALE): Solving Tensor Network Structure Search with Fewer Evaluationscited by 12
  6. 2023Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representationcited by 11
  7. 2025Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEscited by 11
  8. 2024A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problemscited by 6
  9. 2022Cost-efficient Gaussian Tensor Network Embeddings for Tensor-structured Inputscited by 5
  10. 2023From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEscited by 5
  11. 2024tnGPS: Discovering Unknown Tensor Network Structure Search Algorithms via Large Language Models (LLMs)cited by 5
  12. 2021A block-sparse Tensor Train Format for sample-efficient high-dimensional Polynomial Regressioncited by 3
5 more
  1. 2022Position: Tensor Networks are a Valuable Asset for Green AIcited by 3
  2. 2025Tensor Gaussian Processes: Efficient Solvers for Nonlinear PDEscited by 1
  3. 2024Sampling from Boltzmann densities with physics informed low-rank formatscited by 0
  4. 2026Tensor Train Diffusion: Leveraging Low-Rank Structures for High-Dimensional Score-Based Samplingcited by 0
  5. 2026Accelerated Learning of High Dimensional Functions with a Tensor-Featured Training Networkcited by 0