MMGP - Mesh Morphing Gaussian Process - A Mesh Morphing Gaussian Process-Based Machine Learning Method for Regression of Physical Problems

Aug 5, 2026

Learning simulations for industrial design rely on predicting physical phenomena under geometrical variability. Classical regression works well for parameterized shapes, but real inference scenarios may lack shape parametrization and provide only mesh discretizations, which makes learning difficult and often motivates deep graph neural networks. This work introduces MMGP, a Gaussian-process-based approach using mesh morphing onto a common support, dimensionality reduction, and finite element interpolation to handle non-parameterized geometric changes, provide predictive uncertainties, and remain competitive in accuracy and training efficiency, including on large meshes.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages27

Reading Time1 h 8 min

AccessFree to download

Abstract

1 Introduction

2 Preliminaries and related works

3 Methodology

4 Numerical experiments

5 Conclusion

arXiv :2305 . 12871v2 [ cs .LG] 22 Oct 2023 MMGP: A MESH MORPHING GAUSSIAN PROCESS-BASED MACHINE LEARNING METHOD FOR REGRESSION OF PHYSICAL PROBLEMS UNDER NON-PARAMETERIZED GEOMETRICAL VARIABILITY Fabien Casenave, Brian Staber and Xavier Roynard Safran Tech, Digital Sciences & Technologies 78114 Magny-Les-Hameaux, France {fabien.casenave, brian.staber, [xavier.roynard}@safrangroup.com](xavier.roynard}@safrangroup.com) ABSTRACT When learning simulations for modeling physical phenomena in industrial designs, geometrical variabilities are of prime interest. While classical regression techniques prove effective for parameterized geometries, practical scenarios often involve the absence of shape parametrization during the inference stage, leaving us with only mesh discretizations as available data. Learning simulations from such mesh-based representations poses significant challenges, with recent advances relying heavily on deep graph neural networks to overcome the limitations of conventional machine learning approaches. Despite their promising results, graph neural networks exhibit certain drawbacks, including their dependency on extensive datasets and limitations in providing built-in predictive uncertainties or handling large meshes. In this work, we propose a machine learning method that do not rely on graph neural networks. Complex geometrical shapes and variations with fixed topology are dealt with using well-known mesh morphing onto a common support, combined with classical dimensionality reduction techniques and Gaussian processes. The proposed methodology can easily deal with large meshes without the need for explicit shape parameterization and provides crucial predictive uncertainties, which are essential for informed decision-making. In the considered numerical experiments, the proposed method is competitive with respect to existing graph neural networks, regarding training efficiency and accuracy of the predictions. 1 Introduction Many problems in science and engineering require solving complex boundary value problems. Most of the time, we are interested in solving a partial differential equation (PDE) for multiple values of input parameters such as material properties, boundary conditions, initial conditions, or geometrical parameters. Traditional numerical methods such as the finite element method, finite volume method, and finite differences require fine discretization of time and space in order to be accurate. As a result, these methods are often computationally expensive, especially when the boundary value problem needs to be repeatedly solved for extensive exploration of the input parameters space. To overcome this issue, machine and deep learning have been leveraged for various tasks in computational physics, namely, solving and learning solutions to PDEs [37, 56, 65, 81], accelerating linear solvers [5, 34], reducedorder modeling [49], domain decomposition [41], closure modeling [52], and topology optimization [74], to name a few. As reported in the review papers [13, 17, 75], most of the recent advances have been relying on deep neural networks for their flexibility and expressiveness. In this work, we focus on learning simulations of physical phenomena, that are discretized on a non-parameterized unstructured MMGP: Mesh Morphing Gaussian Process method for regression of physical problems mesh. In this situation, traditional machine learning approaches cannot easily be leveraged as the inputs of the problem are given by graphs with different numbers of nodes and edges. In contrast, deep learning models such as graph neural networks (GNNs) [67] can easily overcome this limitation thanks to their ability to operate on meshes with different resolutions and topologies. While GNNs show promising results and their flexibility is highly appealing, they still suffer from a few shortcomings that prevent their deployement in engineering fields where decisions involve high stakes. Training GNNs

FAQs

What challenge does MMGP address in regression of physical problems?

MMGP targets cases where geometrical variability is non-parameterized and only mesh discretizations are available at inference time, making standard machine learning inputs inconsistent across meshes.

How does MMGP avoid relying on graph neural networks?

It uses mesh morphing onto a common support, classical dimensionality reduction, and Gaussian processes combined with finite element interpolation rather than operating directly on graph structures.

What key benefit does MMGP provide compared with deep learning approaches?

MMGP readily delivers predictive uncertainties, supports training efficiently on CPU hardware, and can handle large meshes without explicit shape parametrization.

Aug 5, 2026

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