mlsanityData to 7 Oct 2026

Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems

arXiv 2112.03749 ↗First posted 7 Dec 2021math.NA cs.NA math.PRcited by 21

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Abstract (arXiv)

Solving high-dimensional partial differential equations is a recurrent challenge in economics, science and engineering. In recent years, a great number of computational approaches have been developed, most of them relying on a combination of Monte Carlo sampling and deep learning based approximation. For elliptic and parabolic problems, existing methods can broadly be classified into those resting on reformulations in terms of textitbackward stochastic differential equations (BSDEs) and those aiming to minimize a regression-type L2-error (textitphysics-informed neural networks, PINNs). In this paper, we review the literature and suggest a methodology based on the novel textitdiffusion loss that interpolates between BSDEs and PINNs. Our contribution opens the door towards a unified understanding of numerical approaches for high-dimensional PDEs, as well as for implementations that combine the strengths of BSDEs and PINNs. The diffusion loss furthermore bears close similarities to textit(least squares) temporal difference objectives found in reinforcement learning. We also discuss eigenvalue problems and perform extensive numerical studies, including calculations of the ground state for nonlinear Schr"odinger operators and committor functions relevant in molecular dynamics.

Key facts. Cited by 21 papers in the map (data through 7 Oct 2026). Builds on 42 of its 87 references that are in the map. First posted on arXiv 7 Dec 2021.

Builds on

42 of 87 references.

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30 more
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Citing papers per year: 2022: 2; 2023: 5; 2024: 3; 2025: 5; 2026: 6.

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  1. 2022An optimal control perspective on diffusion-based generative modelingcited by 145
  2. 2023Transport meets Variational Inference: Controlled Monte Carlo Diffusionscited by 21
  3. 2024Dynamical Measure Transport and Neural PDE Solvers for Samplingcited by 19
  4. 2025From discrete-time policies to continuous-time diffusion samplers: Asymptotic equivalences and faster trainingcited by 17
  5. 2022Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learningcited by 13
  6. 2023Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representationcited by 11
  7. 2024Solving Poisson Equations using Neural Walk-on-Spherescited by 11
  8. 2025Adaptive Destruction Processes for Diffusion Samplerscited by 6
  9. 2023From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEscited by 5
  10. 2025A brief review of the Deep BSDE method for solving high-dimensional partial differential equationscited by 5
  11. 2025Estimating Committor Functions via Deep Adaptive Sampling on Rare Transition Pathscited by 3
  12. 2025Integration Matters for Learning PDEs with Backward SDEscited by 3
9 more
  1. 2023Deep Stochastic Mechanicscited by 2
  2. 2023Score Operator Newton transportcited by 1
  3. 2024Control, Transport and Sampling: Towards Better Loss Designcited by 0
  4. 2026Deep Neural networks for solving high-dimensional parabolic partial differential equationscited by 0
  5. 2026A Schr"odinger Eigenfunction Method for Long-Horizon Stochastic Optimal Controlcited by 0
  6. 2026Unbiased and Second-Order-Free Training for High-Dimensional PDEscited by 0
  7. 2026Feature Learning for the High Dimensional Stationary Sch"odinger Equation with Deep Ritz Methodcited by 0
  8. 2026Tensor Train Diffusion: Leveraging Low-Rank Structures for High-Dimensional Score-Based Samplingcited by 0
  9. 2026Forward Trajectory Steering for Hamilton-Jacobi Reachability Analysiscited by 0