mlsanityData to 7 Oct 2026

Physics-informed neural networks for operator equations with stochastic data

arXiv 2211.10344 ↗First posted 15 Nov 2022cs.LG cs.NA math.NAcited by 1

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Abstract (arXiv)

We consider the computation of statistical moments to operator equations with stochastic data. We remark that application of PINNs -- referred to as TPINNs -- allows to solve the induced tensor operator equations under minimal changes of existing PINNs code, and enabling handling of non-linear and time-dependent operators. We propose two types of architectures, referred to as vanilla and multi-output TPINNs, and investigate their benefits and limitations. Exhaustive numerical experiments are performed; demonstrating applicability and performance; raising a variety of new promising research avenues.

Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Builds on 16 of its 43 references that are in the map. First posted on arXiv 15 Nov 2022.

Builds on

16 of 43 references.

References · 16

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2019DeepXDE: A deep learning library for solving differential equationscited by 617
  3. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  4. 2021Physics-informed neural networks with hard constraints for inverse designcited by 277
  5. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  6. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  7. 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
  8. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  9. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  10. 2018Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantificationcited by 81
  11. 2019Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networkscited by 63
  12. 2022NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operatorscited by 33
4 more
  1. 2022Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problemscited by 18
  2. 2022Multi-Output Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Uncertaintiescited by 14
  3. 2021Deep neural network surrogates for non-smooth quantities of interest in shape uncertainty quantificationcited by 4
  4. 2020A matrix-free isogeometric Galerkin method for Karhunen-Lo`eve approximation of random fields using tensor product splines, tensor contraction and interpolation based quadraturecited by 1

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Citing papers per year: 2023: 1.

1 paper.

  1. 2023h-analysis and data-parallel physics-informed neural networkscited by 4