Generative Models for Parameter Space Reduction applied to Reduced Order Modelling

2025/06/11 by Guglielmo Padula, Gianluigi Rozza, Padula, Guglielmo +1
Computer Science · Materials Science · Physics and Astronomy · #FOS: Mathematics #Machine Learning in Materials Science #Model Reduction and Neural Networks #Neural Networks and Reservoir Computing #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.2506.09721

openalex publication_date 2025/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Solving and optimising Partial Differential Equations (PDEs) in geometrically parameterised domains often requires iterative methods, leading to high computational and time complexities. One potential solution is to learn a direct mapping from the parameters to the PDE solution. Two prominent methods for this are Data-driven Non-Intrusive Reduced Order Models (DROMs) and Parametrised Physics Informed Neural Networks (PPINNs). However, their accuracy tends to degrade as the number of geometric parameters increases. To address this, we propose adopting Generative Models to create new geometries, effectively reducing the number of parameters, and improving the performance of DROMs and PPINNs. The first section briefly reviews the general theory of Generative Models and provides some examples, whereas the second focusses on their application to geometries with fixed or variable points, emphasising their integration with DROMs and PPINNs. DROMs trained on geometries generated by these models demonstrate enhanced accuracy due to reduced parameter dimensionality. For PPINNs, we introduce a methodology that leverages Generative Models to reduce the parameter dimensions and improve convergence. This approach is tested on a Poisson equation defined over deformed Stanford Bunny domains.

Citations

• Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connections
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• On choosing and bounding probability metrics

Related

• Reduced Order Modeling of Partial Differential Equations on Parameter-Dependent Domains Using Deep Neural Networks
• Generative Reduced Basis Method
• Reduced-order modeling for parameterized PDEs via implicit neural representations
• LatentPINNs: Generative physics-informed neural networks via a latent representation learning
• Handling geometrical variability in nonlinear reduced order modeling through Continuous Geometry-Aware DL-ROMs