mlsanityData to 14 Oct 2026

Random Grid Neural Processes for Parametric Partial Differential Equations

arXiv 2301.11040 ↗First posted 26 Jan 2023cs.LG cs.NA math.NAcited by 8

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Abstract (arXiv)

We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning probability measures to the spatial domain, which allows us to treat collocation grids probabilistically as random variables to be marginalised out. Adapting this spatial statistics view, we solve forward and inverse problems for parametric PDEs in a way that leads to the construction of Gaussian process models of solution fields. The implementation of these random grids poses a unique set of challenges for inverse physics informed deep learning frameworks and we propose a new architecture called Grid Invariant Convolutional Networks (GICNets) to overcome these challenges. We further show how to incorporate noisy data in a principled manner into our physics informed model to improve predictions for problems where data may be available but whose measurement location does not coincide with any fixed mesh or grid. The proposed method is tested on a nonlinear Poisson problem, Burgers equation, and Navier-Stokes equations, and we provide extensive numerical comparisons. We demonstrate significant computational advantages over current physics informed neural learning methods for parametric PDEs while improving the predictive capabilities and flexibility of these models.

Key facts. Cited by 8 papers in the map (data through 14 Oct 2026). Builds on 39 of its 69 references that are in the map. First posted on arXiv 26 Jan 2023.

Builds on

39 of 69 references.

References · 39

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27 more
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Built on by

Cited by year

Citing papers per year: 2023: 1; 2024: 6; 2025: 1.

8 papers.

  1. 2024Dual Cone Gradient Descent for Training Physics-Informed Neural Networkscited by 21
  2. 2024Generating synthetic data for neural operatorscited by 11
  3. 2024Efficient Prior Calibration From Indirect Datacited by 11
  4. 2024Physics-Aware Neural Implicit Solvers for multiscale, parametric PDEs with applications in heterogeneous mediacited by 5
  5. 2025DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modelingcited by 5
  6. 2024A Primer on Variational Inference for Physics-Informed Deep Generative Modellingcited by 3
  7. 2024R'enyi Neural Processescited by 1
  8. 2023Ensemble models outperform single model uncertainties and predictions for operator-learning of hypersonic flowscited by 0