| Abstract: |
| This presentation highlights recent advancements in reduced order modeling (ROM) and physics-informed neural networks (PINNs) for solving large scale partial differential equations (PDEs). Specifically, we compare several projection-based reduced order models (PROMs) with traditional finite element solvers on elliptic and parabolic problems, where the PDEs are first discretized and parameterized into a full order linear finite element system using NumPy/SciPy packages. The full order model is then projected onto a reduced space spanned by a reduced basis, which is constructed using strong greedy algorithms, proper orthogonal decomposition, and weak greedy algorithms. The coupled computation framework utilizes the discretization and reduced basis construction in the open-source pyMOR package, integrated with the finite element solver package FEniCS. Additionally, we apply physics-informed neural networks to incompressible and compressible Navier-Stokes equations to compare them with direct numerical solvers for both forward and inverse problems. Coarse grid solutions generated by numerical solvers are used to train and test PINNs using the open-source DeepXDE package with multiple parameter setups. Finally, we demonstrate the advantages and limitations of each ROM and PINNs approach under different conditions to illustrate their efficiency and accuracy in solving large-scale PDEs. |
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