Deep learning enhanced model order reduction for parametric problems with discontinuities
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Classical numerical solvers are unscalable for parametric PDEs in a many-query context. They are easily rendered impractical when used for real-world applications. Model order reduction using reduced basis methods can alleviate this issue, although still intrusive in nature. Moreover, methods like proper orthogonal decomposition generate linear sub-spaces and further treatment is required for non-linear parametric problems [1]. In particular, constructing non-linear reduced order models, which can capture discontinuities as well, is a challenging task. Thus, a requirement for non-linear reduced order models with capability to handle discontinuities arises. This contribution addresses these issues, by using deep learning methods, which can generate reduced order models that not only can capture the non-linearities but also can be queried numerous times within few seconds. An autoencoder coupled with an additional network, can predict full order solutions for new unseen parameter sets [2]. The method is benchmarked using a parametric fracture mechanics problem discretized using splines [3]. The results show a substantial reduction in the time required to predict new full order solutions with a negligible compromise in accuracy. References: [1] Chasapi, Margarita, Pablo Antolin, and Annalisa Buffa. "A localized reduced basis approach for unfitted domain methods on parameterized geometries." Computer Methods in Applied Mechanics and Engineering 410 (2023): 115997. [2] Franco, Nicola, Andrea Manzoni, and Paolo Zunino. "A deep learning approach to reduced order modelling of parameter dependent partial differential equations." Mathematics of Computation 92.340 (2023): 483-524. [3] De Luycker, Emmanuel, et al. "X‐FEM in isogeometric analysis for linear fracture mechanics." International Journal for Numerical Methods in Engineering 87.6 (2011): 541-565.