A deep learning solution approach for high-dimensional random differential equations

Mohammad Amin Nabian, Hadi Meidani

Research output: Contribution to journalArticlepeer-review


Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution approach for these problems based on deep learning. This approach is intrusive, entirely unsupervised, and mesh-free. Specifically, the random PDE is approximated by a feed-forward fully-connected deep residual network, with either strong or weak enforcement of initial and boundary constraints. Parameters of the approximating deep neural network are determined iteratively using variants of the Stochastic Gradient Descent (SGD) algorithm. The satisfactory accuracy of the proposed approach is numerically demonstrated on diffusion and heat conduction problems, in comparison with the converged Monte Carlo-based finite element results.

Original languageEnglish (US)
Pages (from-to)14-25
Number of pages12
JournalProbabilistic Engineering Mechanics
StatePublished - Jul 2019


  • Curse of dimensionality
  • Deep learning
  • Deep neural networks
  • Least squares
  • Random differential equations
  • Residual networks

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Civil and Structural Engineering
  • Nuclear Energy and Engineering
  • Condensed Matter Physics
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering


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